Evaluations of annular Khovanov–Rozansky homology

We describe the universal target of annular Khovanov–Rozansky link homology functors as the homotopy category of a free symmetric monoidal linear category generated by one object and one endomorphism. This categorifies the ring of symmetric functions and admits categorical analogues of plethystic transformations, which we use to characterize the annular invariants of Coxeter braids. Further, we prove the existence of symmetric group actions on the Khovanov–Rozansky invariants of cabled tangles and we introduce spectral sequences that aid in computing the homologies of generalized Hopf links. Finally, we conjecture a characterization of the horizontal traces of Rouquier complexes of Coxeter braids in other types.

The positive part of the HOMFLY-PT skein algebra of the annulus is defined as a linear span of annular closures of braids, modulo certain skein relations. A classical result of Turaev [Tur88] states that this skein algebra is isomorphic to the algebra Λ q of symmetric functions in infinitely many variables over [q ±1 ]. In particular, to any braid one can associate a symmetric function which is invariant under conjugation of the braid.
Conversely, many interesting symmetric functions and relationships between them can be represented in terms of (colored) braid closures. For example, if Schur functions correspond to the colored unknots, then certain "plethystically transformed" skew Schur functions s λ/µ [X(q − q −1 )] are represented by "Coxeter braids" (see Section 2 for precise definitions).
Furthermore, the skein of the annulus acts on the (relative) skein of the disk. In particular, after an extension of scalars, there is a homomorphism of Λ q to the Hecke algebra H n for any n, and its image coincides with the center of H n . (1) The motivation for this paper is to study lifts of this homomorphism to the categorified level.
1.1. The annular category. In a series of recent papers [QR18,QRS18] Queffelec, Rose and Sartori categorified the skein of the annulus using annular Khovanov-Rozansky homology. The target for this annular link homology functor is a monoidal category whose objects are (complexes of) oriented webs in the annulus, and the morphisms are given by annular foams. They prove that this category is generated by collections of k -colored essential unknots, and provide an explicit algorithm of simplification of a given web to this basis. The monoidal structure is given by placing one annulus inside another. We reformulate their result and prove the following: Theorem 1.1. The Karoubi completion (or bounded homotopy category) of the category of positive annular webs and foams is equivalent to (the bounded homotopy category of ) the free symmetric monoidal graded Karoubian categoryP generated by a single object E (corresponding to the uncolored essential circle) with an endomorphism x ∈ End(E) (corresponding to a dotted cylinder on the circle) of degree two. Under this equivalence, the k -colored unknot corresponds to the antisymmetric component in E ⊗k .
In other words, the target of the annular Khovanov-Rozansky invariant can be thought of as a category of complexes of Schur functors of E, which categorify the corresponding symmetric functions in Λ q . We will call the bounded homotopy category K b (P) the annular category.
Remark 1.2. It is important to mention that we work in characteristic zero, where the representation theory of S n is semisimple, and Schur functors are well-defined. In finite characteristic, one may need to use the formalism of strict polynomial functors [FS97,HY13b,HY13a,HTY14], but we do not pursue it in this paper.
It is conjectured [QR18,Conjecture 5.4] that every annular web is actually isomorphic to a direct sum of collections of k -colored essential unknots (that is, to a complex concentrated in one homological degree). Here we prove that at least after Karoubi completion this is indeed the case: 1.3. Positive Coxeter braids. Next, we describe another natural generating set in the annular category, which appears in the image of annular Khovanov-Rozansky functors, namely the images of closures of Coxeter braids.
We describe the differential in this complex explicitly. We also describe the spaces of morphisms between various products of C + n . Theorem 1.12. We have End(C + n ) = (ξ 1 , . . . , ξ n−1 ) ⊗ [x], where ξ i are odd variables of homological degree −1 and q-degree 2i − 2 and x has q-degree 2.
Moreover, there are natural "merge" and "split maps" M m,n : C + m ⊗ C + n → qC + m+n [1], S m,n : C + m+n → qC + m ⊗ C + n . and we expect that all morphisms between tensor products of C + n are generated by these and the action of ξ i and x.
More precisely (see Section 5.3 for details) for ribbon skew shapes there exists a canonical left ideal V ǫ ⊂ [S n ] with Frobenius character s ν(ǫ) , and [S n ] ∼ = ⊕ ǫ V ǫ . We let p ǫ ∈ [S n ] denote the idempotent projecting to V ǫ . Theorem 1.13 (Theorem 5.25). Let U n = Span(x i − x i+1 ) be the (n − 1)-dimensional reflection representation of S n , there is a natural S n -equivariant map D : U n ⊗ E ⊗n → E ⊗n . Consider the Koszul complex Cube n := ( • U n ⊗ E n , D).
Example 1.14. For ǫ = (+1, · · · , +1) the skew shape ν(ǫ) has one row, so s ν(ǫ) = s n , the corresponding representation V ǫ is trivial, and the corresponding projector p ǫ is the symmetrizer. Therefore p ǫ · Cube n = (Cube n ) Sn . In Lemma 3.20 we check that this indeed agrees, up to a homological shift, with the description of the annular complex for the positive Coxeter lift in Theorem 1.11, and yields immediately the differentials in it.
By a result of Solomon, the analogues of the projectors p ǫ can be defined for all finite Coxeter groups. We conjecture that Theorem 1.13 can be generalized too, see Conjecture 7.4. 1.5. Organization of the paper. In Section 2 we list various important results about the skein algebra of the annulus, following Turaev [Tur88], Aiston and Morton [AM98,Mor02]. We identify this skein with the algebra of symmetric functions in infinitely many variables, and identify certain closed braids with explicit symmetric functions. In particular, we prove Theorem 2.20 which is a decategorified version of Theorem 1.13. In Section 3 we use Schur functors in symmetric monoidal categories to describe an explicit categorification of the algebra of symmetric functions and a plethystic transformation. In Section 4 we define and study the category of webs and foams and the corresponding Khovanov-Rozansky functor. We prove Theorems 1.1 and 1.3.
In Section 5 we identify the annular complexes for all lifts of the Coxeter element to the braid group and prove Theorems 1.11 and 1.13.
In Section 6 we describe the operation of "wrapping" an annular link around a braid, and prove Theorem 1.9. In Section 7 we briefly discuss a conjectural description of annular homology (or, rather, a class in the horizontal trace) for Coxeter lifts outside of type A. Finally, in the appendix we list some useful facts from homological algebra, in particular, on splitting of homotopy idempotents and triangulated Karoubian categories.

The classical story
In this section we recall the classical constructions related to the skein algebra of the annulus.
2.1. The skein of the annulus. Let A denote an annulus on the plane. The closure of a braid is a link in A × [0, 1]. We define the positive part of the skein of the annulus Sk + (A) as the [q ±1 ]-linear span of all braid closures, considered up to regular isotopy, modulo the HOMFLY skein relation: This can be given an algebra structure by stacking (A × [0, 1]) ⊔ (A × [1, 2]) = A × [0, 2]. We will refer to this operation as to skein product, which should not be confused with the product of braids. The skein product of two braid closures is isotopic to the disjoint union of the two braid closures, considered as living in two annuli, one outside of another. An Eckmann-Hilton argument then implies that Sk + (A) is a commutative algebra with respect to the skein product.
Theorem 2.1. [Tur88] The skein algebra Sk + (A) is isomorphic to the algebra Λ q of symmetric functions in infinitely many variables over [q ±1 ].
There are several versions of the isomorphism in Theorem 2.1 which differ by automorphisms of the symmetric function ring, possibly after extending scalars. We outline one of them in the next section.
2.2. Universal Hecke trace and symmetric functions. The Hecke algebra H n is defined as the quotient of [q ±1 ]Br n by the HOMFLY skein relation shown above. It is easy to see that H n is spanned by the images of positive permutation braids. Moreover, taking braid closures in the annulus defines a linear map Tr : H n → Sk + (A). Since the closures of conjugate braids represent the same link in the annulus, we have Tr(ab) = Tr(ba). In fact, it is easy to see from the construction that In other words, any linear map f : H n → V such that f (ab) = f (ba) factors through the map Tr : H n → Sk + (A).
The identification of Sk + (A) with Λ q is also transparent in this construction. Indeed, the irreducible representations V λ of H n are classified by Young diagrams λ with n boxes. Define the map Tr Λq : H n → Λ q , Tr Λq (x) = λ Tr(x, V λ )s λ where s λ is the Schur function. Clearly, Tr Λq (ab) = Tr Λq (ba), so by the above Tr Λq factors through Sk + (A): (2) Tr Λq (x) = i(Tr(x)), i : Sk + (A) → Λ q .
Theorem 2.1 states that i is an isomorphism.
Remark 2.2. The Hecke algebra can also be used to study invariants of oriented tangles with n inputs and n outputs. More precisely, we consider the ring := [q ±1 , a ±1 , (q k − q −k ) −1 ] for all k > 1, and the -module Sk(n, n) spanned by all framed oriented tangles in an axis-parallel rectangle in Ê 2 , with n inputs on the bottom boundary and n outputs on the top, modulo the HOMFLY skein relation and: It is known [AM98,MT90] that Sk(n, n) (with respect to composition) is isomorphic to the Hecke algebra H n ⊗ with scalars extended to . The extended trace is denoted by Tr Λa,q : Sk(n, n) → Λ q ⊗ =: Λ a,q .
The universal trace can be specialized to the Jones-Ocneanu trace on the Hecke algebra which yields the HOMFLY-PT polynomial or sl N Reshetikhin-Turaev invariants of links L ⊂ S 3 presented as braid closures.
Proposition 2.3. Let f L ∈ Λ q correspond to a braid closure L in the thickened annulus under the isomorphism (2). Then the sl N Reshetikhin-Turaev invariants L N and the HOMFLY-PT polynomial L can be computed as follows: Here p k denotes the k-th power sum symmetric function.
Proof. Part (a) is well-known. To obtain (b), observe that by (a) 2.3. Coxeter braids. In this section we compute the images of braid lifts of Coxeter elements in Λ q . To this end, we introduce a particular plethysm operation. Recall that the power sum symmetric functions p n for n > 1 give an algebraically independent set of generators of Λ q ⊗ É.
Lemma 2.4. There exists a unique [q ±1 ]-algebra endomorphism of Λ q which sends p k to its scalar multiple p k (q −k − q k ) for all k ≥ 1.
If f ∈ Λ q is a symmetric function, we denote its image under this endomorphism by f [X(q −1 − q)].
Proof. After extending to scalars to É[q ±1 ], it is clear that there is a unique endomorphism with these properties. The fact that it is well-defined over [q ±1 ] follows from the following lemma, which can be used to compute the images of the algebraically independent integral generators given by the elementary (or complete) symmetric functions e n (or h n ) for n ≥ 0.
Lemma 2.5. We have Proof. Consider the identity of generating functions By taking the coefficient at z n , we get The other identity admits an analogous proof.
The following proposition describes the trace of the positive Coxeter braid σ n−1 · · · σ 1 in terms of the plethysm operation introduced above.
Proof. The traces of σ n−1 · · · σ 1 in various representations of the Hecke algebra can be found in [Jon87, Section 9]. Such a trace in V λ vanishes if λ is not a hook, and equals (−1) i q n−1−2i for the hook λ i = (n − i, 1 i ). It remains to apply Lemma 2.5.
Remark 2.8. In [Tur88] Turaev identified the entire skein of the annulus Sk(A) with a polynomial algebra in variables l k with k ∈ \{0}. These l k can be chosen to be the images of the closures of positive Coxeter braids on |k| strands, winding positively or negatively around the annulus. The positive half has generators l k for k ≥ 1 and is thus isomorphic to the ring of symmetric functions.
Definition 2.10. Let a be a composition of n. We define a symmetric function where h k are complete symmetric functions and l(a), l(b) are the lengths of a and b as above.
Example 2.13. In our running example we get Proof. Follows from the determinantal formula for Ψ(a) and the Jacobi-Trudy formula for s k+1,1 n−k−1 .
For general a, the Schur expansion for Ψ(a) is more complicated.
We are ready to connect these combinatorial results to knot theory.
In the following, we use the notation c a for the annular closure of σ ǫ and |a| + := |ǫ| + .
Proof. Let us prove the statement by induction on the number of entries −1 in ǫ. If ǫ = (+1, · · · , +1), this follows from Proposition 2.6. Otherwise, let a = (a 1 , . . . , a s ) be the corresponding composition. The rightmost negative crossing in σ ǫ is at position a 1 + . . . + a s−1 . If we replace it by a positive one, we get the composition a ′ = (a 1 , . . . , a s−2 , a s−1 + a s ). If we erase that crossing, we get a disjoint union of a Coxeter braid for the composition a ′′ = (a 1 , . . . , a s−1 ) and a positive Coxeter braid on a s strands. Now, by the skein relation the we get The last equation follows from the recursive formula (4).
Proof. Follows from Theorem 2.20 and Corollary 2.14.
Corollary 2.22. We have Proof. This follows from Corollary 2.21 and the equations Remark 2.23. This corollary was proved by Aiston [Ais97] by different methods, see also [Mor02].
2.4. From skein to the center of Hecke algebra. The skein of the annulus is closely related to the center of the Hecke algebra, as exemplified by Morton [Mor02]. Recall that the Jucys-Murphy braids are defined as L i = σ i−1 · · · σ 1 σ 1 · · · σ i−1 . It is easy to see that L i L j = L j L i for all 1 ≤ i, j ≤ n. Note that L 1 is a trivial braid. It is well known that the center of H n is spanned by the symmetric polynomials in L 1 , . . . , L n . There is a natural homomorphism T n from Sk + (A) to H n ⊗ ∼ = Sk(n, n) given by wrapping annular links L around the identity braid on n strands as in the following picture: It is easy to see that for any annular link L the tangle T n (L) is central in Sk(n, n) (and hence in H n ⊗ ), and T n (L 1 ⊔ L 2 ) = T n (L 1 )T n (L 2 ).
Theorem 2.24. [Mor02, Theorem 3.9] Under the identification Sk + (A) ⊗ ∼ = Λ a,q one has T n (f ) = φ n (f )(L 1 , . . . , L n ), where f ∈ Λ q and φ : Λ a,q → Λ a,q is an endomorphism defined by where ε is the evaluation homomorphism defined in Proposition 2.3. 1 It is sometimes helpful to rewrite (6) in terms of the eigenvalues of central elements T n (f ). Recall that L i can be simultaneously diagonalized using Jones-Wenzl-type projectors. For each standard Young tableau T there is an element p T ∈ Sk(n, n) such that L i · p T = q 2ci(T ) p T , where c i denotes the content of the box labeled by i in T , see e.g.  Lemma 2.25. Assume that λ has at most N parts. Given a symmetric function f ∈ Λ q and a standard tableau T of shape λ, one has Proof. Since p T is an eigenvector for all L i , by Theorem 6 it is an eigenvector for T n (f ) for any f , so for some scalar µ T (f ). Clearly, the assignment f → µ T (f ) is a ring homomorphism, so it is sufficient to compute the image of power sums. We have so the eigenvalue of (L k 1 + . . . By applying (6) we get , . . . , q −2λN +(N −1) ).
2.5. Generalized Hopf links. We can use the above results to describe the polynomial invariants of generalized Hopf links. Consider the standard genus one Heegaard decomposition of S 3 with two annular links L 1 , L 2 in the two genus one handlebodies. Their union H(L 1 , L 2 ) is a link in S 3 which we call a generalized Hopf link (indeed, the cores of the two solid tori yield a Hopf link). Note that it is naturally framed by framings of L 1 and L 2 . The following is clear from the definition: Proposition 2.26. The sl N polynomial H(L 1 , L 2 ) N depends only on classes of L 1 and L 2 in Sk + (A), and it is bilinear in these classes.
To compute this invariant, it is then sufficient to choose a basis in Sk + (A) ≃ Λ q and to compute the bilinear form in this basis. Lemma 2.25 immediately implies the following: Proposition 2.27. The invariants of the generalized Hopf links are completely determined by either of the following: (a) If both components are colored by Schur functions then  Remark 2.28. It follows that the right hand side of (7) is symmetric in λ and µ for all N .
3. General facts about symmetric monoidal categories 3.1. A free symmetric monoidal category. We start by defining a useful PROP -a graded, additive version of a product and permutations category [Mac65, Chapter V, 2.4].
Definition 3.1. Let P denote the graded, strict symmetric monoidal -linear additive category that is freely generated by a single object E and a degree two endomorphism x. We will use the notation P = Kar(P) for its idempotent completion.
The objects of P are formal direct sums of grading shifts of tensor powers of E and we denote such grading shifts by powers of q. The morphisms of P are matrices whose entries can be interpreted aslinear combinations of string diagrams built from identity endomorphisms of copies of E, the morphism x : q k E → q k−2 E and the basic braiding morphism σ : q k E ⊗ E → q k E ⊗ E. (We think of such string diagrams as dotted permutations). Explicitly, we have: Here, the subscript k − l indicates taking the degree k − l component of this algebra, which is graded by putting all x i in degree two and all permutations in degree zero. In other words, we have: In the following K 0 (C) denotes the split Grothendieck group (ring) of an additive (monoidal) category C and Λ q is the [q ±1 ]-algebra of symmetric functions in infinitely many variables. Proof. By definition, P is additively generated by q k E ⊗n and there are no isomorphisms between distinct such objects, so K 0 (P) ∼ = [q ±1 , e] and [E ⊗k ] = e k .
To compute the Grothendieck group ofP, we need to classify the idempotent endomorphisms of objects of the form q k E ⊗n in P. Since x has positive degree, (9) implies that idempotents appear only in [S n ] and they are exactly the Young idempotents e λ , which are parametrized (up to isomorphism) by Young diagrams λ with |λ| = n. Then K 0 (P) has a basis given by the classes of such pairs (q k E ⊗n , e λ ). The fact that this gives a ring homomorphism follows from the next section.
3.2. Schur functors and evaluation. Let C be a -linear strict symmetric monoidal Karoubian category, and let E be an object in C. For every n ≥ 1 there is an action of S n on E ⊗n given by the permutation of the factors. In other words, we have a homomorphism φ n : S n → End(E ⊗n ). For every partition λ of n we pick the primitive Young idempotent e λ ∈ [S n ] corresponding to a fixed Young tableau of shape λ. Its image φ n (e λ ) is an idempotent endomorphism of E ⊗n . Since C is Karoubian, we can define the Schur functor of E as the image of this idempotent: For more details on Schur functors see [Del02]. We will write n (E) = S (1 n ) (E) and S n (E) = S (n) (E).
Definition 3.4. We say that the object E has rank at most N , if N +1 (E) ∼ = 0.
For example, N is of rank at most N in the symmetric monoidal category of complex vector spaces.
Proposition 3.5. If E is an object of rank at most N and λ is a partition with more than N parts then S λ (E) ∼ = 0.
Proposition 3.6. Let C be a graded, strict symmetric monoidal -linear additive category, and let E be an object in C with an endomorphism X. Then there is a unique braided monoidal -linear additive functor P → C which sends E to E and x to X. If, in addition, C is Karoubian then this functor extends to a functorP → C.
Proof. By the assumptions, there is an action of [X 1 , . . . , X n ] ⋊ [S n ] on E ⊗n , so we can define a monoidal functor P → C sending E ⊗n to E ⊗n . It uniquely extends to the Karoubi completions.
Remark 3.7. More generally, let C be a -linear additive monoidal (but not necessary symmetric) Karoubian category. We shall say that an object E ∈ C with an endomorphism X is self-commuting with symmetry s : E ⊗ E → E ⊗ E if there is an additive monoidal functorP → C sending E to E, σ to s, and x to X.

3.3.
Complexes. The constructions from the previous subsection directly extend to the category Kom(C) of complexes of objects in C and to the homotopy category of complexes K b (C). We will frequently use the following fact which is well-known to experts (e. g. [BS01]). For completeness, we prove it in the appendix as Theorem A.10.
Theorem 3.8. The bounded homotopy category of a Karoubian category is Karoubian.
The category of complexes Kom(C) is symmetric monoidal if the original category C was so. To fix the sign conventions, we define the differential on the tensor product by the equation The braiding Σ on Kom(C) differs from the braiding σ in C by sign placements.
(11) Σ Ai,Bj = (−1) ij σ Ai,Bj This allows one to define arbitrary Schur functors for complexes. One can check that Schur functors of homotopy equivalent complexes are homotopy equivalent, see e.g. Theorem A.5. We refer to the appendix for more details on Schur functors for complexes. Also, we record the following fact which immediately follows from the above discussion.
Proposition 3.9. Let C be a -linear additive monoidal (but not necessary symmetric) Karoubian category, assume E is a self-commuting complex in the bounded homotopy category K b (C). Then the Schur functors S λ (E) are well defined. sends a ⊗ b → (−1) deg(a)−1 a ⊗ b. Indeed, in agreement with (10), the isomorphism is given by the braiding c 23 .
Similarly, one can check that the chain of isomorphisms differs from the composition of the braiding A ⊗ B ∼ = B ⊗ A and the shift [2] by a factor of −1. Therefore the representations of S k on (A ⊗k )[k] and on (A[1]) ⊗k differ by sign, and This shows that the notion of the Schur functor of a complex is sensitive to the parity of homological degrees of its terms.
, where E is in homological degree 1 and F is in degree 0. Then: where S k (E) has homological degree k. However, where S k (F ) has homological degree −k.
Example 3.11. Consider a two-term complex over the category .
We will need the following result: Theorem 3.12. Let P be the full tensor subcategory ofP generated by i (E). Then the bounded homotopy categories of P and ofP are equivalent.
Proof. Since P is a full subcategory ofP, the homotopy category of P is a full subcategory of the homotopy category ofP. Furthermore, K b (P) is dense (in the sense of [Tho97]) in K b (P) since every complex in K b (P) is even isomorphic to a direct summand in a complex in K b (P), i.e. a complex built out of several copies of E ⊗n . Every Schur functor of E is homotopy equivalent to a complex built out of i (E). Indeed, the Schur functor S λ (E) appears as a unique summand in j λj (E) and all other summands are smaller than λ in dominance order, so we can inductively resolve S λ (E) by the products of i (E). This means that K 0 (K b (P)) ∼ = K 0 (K b (P)) and by Theorem A.1 we get K b (P) ≃ K b (P).

Affine extensions and plethysms. Consider a symmetric monoidal Karoubian -linear category C. We define its affine extension C[t] as (the Karoubi completion of) the category with the objects E[t]
where E ranges over objects of C, and the hom spaces have the form In particular, each object E[t] in C[t] has endomorphisms t k for k ≥ 0. The tensor product on C naturally induces a tensor product in C[t]. We define pullback and pushforward functors We assume that C is graded, and t has some nontrivial grading, so that the direct sum in the definition of π * (E) makes sense in an appropriate completion with respect to this grading (we allow infinite direct sums which are finite in each grading).
Clearly, π * is monoidal, and left adjoint to π * . These functors naturally extend to functors between the homotopy categories of complexes of objects in C and C[t], respectively.
Example 3.13. If R is an algebra and C = R − mod, then C[t] ≃ R[t] − mod. The functors π * and π * are given by (derived) restriction and induction functors. In particular, if E is a free R-module then E[t] is a free R[t]-module, and all free R[t]-modules appear this way. Furthermore, the restriction of E[t] to R is isomorphic (as an R-module) to E ⊗ [t], and We now use affine extensions to define functors which model certain plethystic transformations. We define a two-term complex overP[t]: Observe that K(E, x) still has an action of x as an endomorphism of a complex. By Proposition 3.6, we can define an evaluation functor fromP to K b (P[t]) which sends an object F ofP to F (K(E, x)).
Definition 3.14. We define the functor Φ :P → K b (P) as the composite: Example 3.15. Recall that we have K 0 (P) ∼ = Λ q and the functor Φ induces the following map on the level of Grothendieck rings: Note that the first map is a ring homomorphism (induced by a monoidal functor), but the second is not.
The "plethysm" functor Φ can be combined with the evaluation in the following way. Let E be an object in a symmetric monoidal Karoubian category C with an endomorphism X. As above, this data defines a braided monoidal functorP → C which sends E to E and x to X, which can be extended to a functor from K b (P) to K b (C). By the functoriality of affine extension, we can also construct functorŝ . It is easy to see that for any object F ofP these send 3.5. Examples of plethysms. Let us compute the action of Φ on some objects and morphisms.
Example 3.16. We have where the last homotopy equivalence follows from "infinite Gaussian elimination".
Definition 3.17. Let U ≃ n−1 denote the reflection representation of S n . Then we define the Koszul complex From the definition it is immediate that Cube n admits an action of S n , which restricts to the symmetryinduced action S n → End(E ⊗n ) in homological degree zero.
is also a Koszul complex, and as such it can be recovered from its last differential, which is the -linear map by taking the exterior algebra on (E ⊗n ) ⊕n and defining the differential as contraction with S. We can obtain an isomorphic Koszul complex after a change of basis from: Considering this as a complex of [x 1 , . . . , x n ]-modules, we can apply Gaussian elimination along the component −t of the differential to obtain Cube n .
Corollary 3.19. Let Cube λ n denote the chain complex obtained as the image of our chosen Young idempotent e λ of shape λ acting on Cube n . Then we have: Proof. The functor π * commutes with the action of [S n ], so We now describe a categorified version of the identity in Lemma 2.5.
Lemma 3.20. The S n -invariant part of Cube n can be written as Proof. It is well known that the exterior powers of U are irreducible representations of S n labeled by the hook Young diagrams. Then we have Similarly, one can prove the following.
Lemma 3.21. The sign-isotypic component of Cube n can be written as As we will see in Theorem 5.1, the complexes shown in the previous lemmas agrees (up to a homological shift) with the annular invariants of the (n − 1)-fold positively and negatively stabilized unknots respectively.
Next, we consider particular evaluations of these complexes. ].
The shown homotopy equivalence holds in the category of complexes of free [t]-modules. We can write [X, t]/X k as a direct sum of k copies of [t] with the action of X shifting them by one. Then we get the following complex of [t]-modules: Here the horizontal arrows are given by multiplication by t and the diagonal ones correspond to X and hence are multiplications by (±1). Gaussian elimination cancels everything except the top left and bottom right copies of [t], which are then connected by t k . Now by Example 3.11 we have for all n ≥ 1.
Generalizing the previous example, let E be a vector space with the action of a nilpotent operator X with Jordan blocks of size k 1 , . . . , k n . Then we can write ]. Therefore The effect of π * on the terms in the sum can be computed using the previous example.
Example 3.24. Suppose that E is a vector space with an endomorphism X which has two Jordan blocks of sizes k 1 and k 2 . Then S n (K(E, X)) has (n + 1) direct summands: ✿✿✿ ], n 1 + n 2 = n, n 1 , n 2 > 0.
After applying the forgetful functor π * , the latter complexes are isomorphic to their homology which have dimension min(k 1 , k 2 ) both in homological degrees one and zero. Therefore

Khovanov-Rozansky theory
4.1. Webs. The Reshetikhin-Turaev invariants of knots, links and tangles are defined as certain intertwiners of representations of quantum groups. In type A, these intertwiners and the relations satisfied by them can be described by a graphical calculus of webs, see [MOY98,CKM14]. The basic building blocks in the cases of sl N and gl N are the fundamental representations a q N q and their identity endomorphisms, as well as two types of natural intertwiners which are called merge and split respectively: Other intertwiners can be built by taking tensor products and composites of identities, merges and splits, and such composites quickly become linearly dependent. Analogously, complicated webs can be built by gluing together the shown basic pieces, which then satisfy corresponding linear relations. We illustrate a few relations here and refer to [CKM14] for a complete list of web relations for sl N and to [TVW17] for the case of gl N . Theorem 4.2. N Web is equivalent to the full subcategory of representations of U q (gl N ) whose objects are the tensor products of exterior power representations a q N q for 0 ≤ a ≤ N . The equivalence sends the object a := (a 1 , . . . , a m ) to a1 q N q ⊗ · · · ⊗ am q N q . Proof. This is a gl N variant of the main result of [CKM14], see also [QS19,TVW17]. Now let S be an oriented surface of finite type, possibly with marked points on the boundary with a labeling and a choice of inward or outward orientation. We denote by N Web(S) the [q ±1 ]-module spanned by properly embedded webs in S, with boundary matching the data on the marked points, modulo isotopy rel boundary and web relations supported in discs D 2 ⊂ S.
N Web(S) is a version of the gl N skein module of the surface S. Oriented, framed links embedded in S × [0, 1] can be evaluated in N Web(S) by projecting to S (enforcing the blackboard framing) and resolving all crossings into alternating sums of webs according to the following rule.
Negative crossings are resolved using an analogous formula with q inverted. The class in N Web(S) represented by an embedded link is invariant under regular isotopy in S × [0, 1]. Framing changes and fork twists act by powers of q, but all fork slides hold on the nose: 4.2. Foams. We still let S denote an oriented surface of finite type. The [q ±1 ]-module N Web(S) admits a graded, additive, -linear categorification N Foam(S) that is closely related to the canopolis N Foam of gl N foams defined in [ETW18] using the closed foam evaluation formula of Robert-Wagner [RW20]. Here we only describe the essential features of N Foam(S) and comment on the necessary variations relative to N Foam. A direct consequence of the local foam relations in N Foam(S) is that we have explicit isomorphisms between webs, which induce the web relations (13) after passing to the Grothendieck group.
Remark 4.4. The use of foams in the categorification of link and tangle invariants has a long history, starting with Bar-Natan's use of linearized cobordism categories in his description of Khovanov homology [BN05]. Khovanov's categorification of the sl 3 link polynomial [Kho04] is the first one that uses foams with singularities. The matrix factorization categories underlying Khovanov-Rozansky gl N link homologies were given a topological interpretation via foams in [KR07], which was used in a new construction of gl N link homologies by Mackaay-Stošić-Vaz [MSV09]. Blanchet demonstrated that gl 2 foams support a version of Khovanov homology that is functorial under link cobordisms [Bla10]. Better control over gl N foam categories was gained by Lauda-Queffelec-Rose through their connections to categorified quantum groups [LQR15,QR16]. Finally, Robert-Wagner [RW20] found a mathematically rigorous and entirely combinatorial construction of gl N foams, which is the basis for the foam categories used here and in the proof of the functoriality of Khovanov-Rozansky homologies under cobordisms in [ETW18].

Categorical invariants for links in a thickened surface.
It is now a routine task to define a categorical invariant of links (or tangles) in S × [0, 1] (with boundary in ∂(S) × {1/2}) that takes values in K b (N Foam(S)), the homotopy category of chain complexes over N Foam(S). Indeed, for a generic tangle embedding, the natural projection S × [0, 1] → S gives a tangle diagram. The alternating sum in the crossing formula (14) gets lifted to a chain complex and if several crossings occur, the alternating multi-sums become tensor product chain complexes. In fact, there are two natural conventions for the chain complexes that can be associated to a positive 3 uncolored crossing: In both cases the differential is given by an unzip foam. For more details about these Khovanov-Rozansky constructions using foams, see e.g. [ETW18, Section 3.1] and [QW21, Section 4]. up to chain homotopy equivalence. While we favour the framed version T D fr in this paper, we also introduce T D since it is known to admit a functorial assignment of chain maps to tangle cobordisms as we describe next.
Definition 4.6. We denote by STan the category with objects given by tangles that are properly embedded in S × [0, 1] and with morphisms given by isotopy classes of tangle cobordisms embedded in S × [0, 1] 2 . For surfaces without specified boundary points, we also denote STan by SLink. Since T D fr differs from T D only in grading shifts in tensor factors, this implies that − fr can also be equipped with functorial cobordism maps. However, we currently do not know whether there is a unique (or at least a distinguished) way of lifting Theorem 4.7 to the framed setting. Another open question is the following. 4.4. Annular links, webs, and foams. In this section we consider the case S = A := S 1 ×[0, 1] without marked points on the boundary, and fix an orientation of the core circle of A.
We define a monoidal structure on ALink as follows. Given two annular links L 1 and L 2 in The definition of ⊠ on morphisms is analogous. It is a simple exercise to check that this defines a monoidal structure with unit given by the empty link and with unitors and associators given by isotopies. In fact, the existence of "vertical" and "horizontal" isotopies give rise to a (non-symmetric) braiding on ALink.
We say an annular link is consistently oriented if it is given as the closure of a braid with orientation matching the orientation of the core circle. We then denote by ALink + the subcategory of ALink given by consistently oriented links and cobordisms whose time-slices are consistently oriented.
It is clear that two braids closures are isotopic in the annulus (and the corresponding objects in ALink + are isomorphic) if and only if the braids are conjugate.
For consistently oriented annular links, there exists a universal categorified link invariant from which all annular and planar Khovanov-Rozansky homologies can be recovered. In order to describe its target category, we say a web W in A is consistently oriented if the tangent vectors project positively to the core circle.
The subcategory N AFoam + of N AFoam is cut out by requiring webs to be consistently oriented and foams to have generic cross-sections that are isotopic to such consistently oriented webs.
We denote by AFoam + the category obtained from N AFoam + by stabilizing N → ∞. In other words, AFoam + is the category of consistently oriented annular webs and foams, without restriction on the labeling set and with a free action of the dot on 1-labeled facets. The component of AFoam + of winding degree n can be identified with the horizontal trace (see Section 7.1) of the category of singular Soergel bimodules of type A n−1 .
Theorem 4.9 ( [QR18]). The annular Khovanov-Rozansky homologies factor through the functor Furthermore, the annular disjoint union yields a natural monoidal structure on AFoam + and its homotopy category, which is respected by the Khovanov-Rozansky functor.
Definition 4.11. We define AFoam + S1 to be the full subcategory of AFoam + whose objects are direct sums of grading shifts of webs that are collections of essential concentric circles in the annulus.
The notation AFoam + S1 is to suggest that the objects in this category are S 1 -equivariant, i.e. that they are invariant under rotation along the core of the annulus. In fact, the same is true for morphisms.
Proposition 4.13 ([QR18, Proposition 5.1]). The inclusion AFoam + S1 ֒→ AFoam + induces an equivalence of categories The main step in the proof of this result is that each annular web, considered as a complex concentrated in homological degree zero, is isomorphic in K b (AFoam + ) to a chain complex built out of concentric circle webs. In fact, this is true more generally, see Proposition 6.6. For now, we take note of the implication that the categorical invariants of braid closures can be assumed to take values in K b (AFoam + S1 ). In the next session we will obtain an alternative description of this category. 4.6. Decorated webs. We can now take quotients of the webs and foams in AFoam + S1 by their free S 1 -symmetry. Under this dimensional reduction, collections of labeled concentric circles are mapped to finite sequences of labeled points on a line Ê. Rotationally symmetric foams are mapped to isotopy classes of webs in the strip Ê × [0, 1], whose edges inherit the decorations by symmetric functions of the foam facets.
Definition 4.14. Let DecWeb denote the non-negatively graded, additive, -linear category of decorated that takes boundary sequences to collections of concentric circles and decorated webs to decorated rotationally symmetric foams.  (18) Lemma 4.17. DecWeb admits a symmetric monoidal structure.
Proof. The tensor product is given by placing webs side by side. The symmetry is an isomorphism of degree zero and given on objects of the form (k, l) by the q = 1 specialization of (14), with a ✿✿✿ sign ✿✿✿✿✿✿✿✿✿ correction: The symmetry on other pairs of objects is constructed as composition of these basic symmetries. For checking the naturality of the symmetry, note that vertices still slide through other strands as in (15) despite the sign correction. It remains to verify that decorations migrate through such crossings. In the case k = l = 1, this follows directly from (19). In the more general case, one first blows up both strands into blisters of parallel 1-labeled strands via relation (13). These blisters fork-slide underneath the crossing, decorations migrate onto the 1-labeled strands by (18) and then through all remaining 1-1-crossings. Then one reverses the process on the other side.
In Theorem 4.19, we will get a more intrinsic characterisation of DecWeb. To prove this theorem, we take a technical detour through modules for Schur quotients of current algebras.  ≥0 . This implies that the system of functors vTr(Φ ∞ ) defines an isomorphism as claimed.
We denote this isomorphism from U to DecWeb again by vTr(Φ ∞ ).
Theorem 4.19. DecWeb is isomorphic to a full subcategory of the symmetric monoidal Karoubian -linear categoryP, which is freely generated by a single object and an endomorphism of degree 2. More specifically, it is isomorphic to the full subcategory P whose objects are tensor products of antisymmetric Schur functors in the generating object.
The following proof is inspired by Cautis-Kamnitzer-Morrison's use of skew Howe duality (a generalisation of Schur-Weyl duality) to describe diagrammatic categories in [CKM14]. For an instance of Schur-Weyl duality for current algebras, see [GKS20, Section 6].
Proof. There is an obvious full, essentially surjective functor Ψ from the said full subcategory P of P to DecWeb, but it remains to show that it is faithful. This will follow from the fact that there is an isomorphism α : P → U such that Ψ = vTr(Φ ∞ ) • α. It suffices to prove this for P, the free symmetric monoidal category on one object E and one endomorphism x (without insisting on any partial idempotent-completeness), and U ′ , the direct limit of idempotent truncations of the form  1 [1,...,1,0 1 [1,...,1,0,...,0] .
We define a functor α : P → U ′ by sending: and then onward to U ′ via the component maps. With this definition of α, we have Ψ = vTr(Φ ∞ ) • α.
A standard argument shows that α is surjective. Namely, a spanning set for morphism spaces in U ′ is given by the images of dotted permutations m≥0 {α(σx n1 1 · · · x nm m )|σ ∈ S m , n i ≥ 0}. It suffices to show that these remain linear independent. To this end, consider U(gl m [t]) as an algebra and the U(gl m [t])module a ( m ⊗ [X]), which decomposes into gl m -weight spaces a1 ( [X]) ⊗ · · · ⊗ am ( [X]). For the weight [1, . . . , 1] we simply get the weight space [X 1 , . . . , X m ]. Since only non-negative weights arise, this descends to a U(gl m [t]) ≥0 -module. It is straightforward to check that pre-composing with α, we obtain the natural action of P where permutations act on indices and x on the i-th strand acts by multiplication by X i . It is then clear that the α-images of dotted permutations act by linearly independent operators, and are thus linearly independent.
Proof. We already know that there exists a fully faithful functorP → Kar(DecWeb) → Kar(AFoam + ) and we shall show that it is essentially surjective. To this end, let W be an annular web. Proposition 4.13 allows us to express W as a chain complex, whose chain groups are collections of concentric circles. After proceeding to the Karoubi envelope, we can decompose these further into Schur functors of a single circle. When considered as a chain complex concentrated in homological degree zero, W is homotopy equivalent to an object C(W ) in K b (P). We may assume this object to be represented by a minimal chain complex. SinceP is non-negatively graded and semi-simple in degree zero, the homotopy equivalence between W and C(W ) is an isomorphism of chain complexes, and thus C(W ) is concentrated in degree zero. This shows that every object in AFoam + is isomorphic to an object inP, and since the latter is idempotent complete by definition, the same holds for every object in Kar(AFoam + ). This verifies essential surjectivity and finishes the proof.
Remark 4.21. It might be helpful to give a more direct explanation why an arbitrary annular web is isomorphic to an object inP, and not just in the homotopy category. Indeed, we can follow the annular simplification algorithm from [QR18] and use bubble removal and square switch relations to reduce a web to a collection of essential circles. At each step of the algorithm, one either replaces a web by an isomorphic one, or presents it as a direct sum of simpler webs, or presents it as a direct summand in a simpler web. SinceP is Karoubian, all these steps show that a web is isomorphic to an object inP, if the simpler webs are.
Note that in [QR18] Queffelec and Rose used a slightly different algorithm where, if a web is presented as a direct summand in a simpler web, it is expressed as a cone of the inclusion of complimentary summands. This way [QR18] avoids Karoubi completion, but steps into the homotopy category. By Theorem 3.12 the two algorithms actually agree in the homotopy category of the Karoubi completionP.

4.7.
Braiding for annular webs. The category of annular links and cobordisms between them has a natural braided monoidal structure. The annular Khovanov-Rozansky functor from this category to the homotopy category of complexes of annular webs and foams preserves the monoidal structure, but a priori it is not clear whether the latter has any braiding. Proof. By Lemma 4.17, DecWeb and thus AFoam + S1 have a symmetric braiding. This immediately extends to K b (AFoam + S1 ). Then we use the equivalence of Proposition 4.13 to transport this symmetric braiding to K b (AFoam + ).
Note that every object W in AFoam + and thus K b (AFoam + ) has a grading [W ] ∈ AE by weighted winding number around the annulus. Besides the braiding σ V,W : V ⊗ W → W ⊗ V on K b (AFoam + ) that was obtained in Proposition 4.22, we will also consider the sign-twisted braiding σ, which is defined by σ V,W = (−1) [V ][W ] σ V,W . Transported back to DecWeb, this braiding is described by the q = 1 specialization of (14), i.e. the formula shown in Lemma 4.17 without ✿✿✿✿ sign ✿✿✿✿✿✿✿✿✿ correction. For the following, let ALink + S1 denote the full subcategory of ALink + with objects being collections of concentric colored circles.
Theorem 4.23. The restricted annular Khovanov-Rozansky functor − : ALink + S1 → K b (AFoam + ) is braided with respect to the standard braiding on K b (AFoam + ). The framed version − fr is braided with respect to the sign-twisted braiding.
Proof. The braiding on ALink + is given by braiding isotopies, i.e. certain cobordisms which braid annular links radially past each other. Under the Khovanov-Rozansky functor − , such maps induce invertible morphisms in K b (AFoam + ), and we shall check that these morphisms agree with the symmetric braiding morphisms in K b (AFoam + ) that were defined in Proposition 4.22. (The case of − fr is analogous and will be omitted.) It suffices to compare these braiding morphisms on pairs of monoidal generators, i.e. two colored circles.
For two uncolored circles, the computation of the maps induced by the braiding cobordism and its inverse is simple-both involve two Reidemeister II moves-and they agree with the braiding from Proposition 4.22. A version of this argument (without foams) appears in Grigsby-Licata-Wehrli [GLW18]. We show the details here for convenience. The braiding of two uncolored circles can be described as a movie of annular link diagrams as follows: This is a composite of a Reidemeister II move, an isotopy of the positive crossing around the annulus, and an inverse Reidemeister II move. In order to compute the composite chain map, we will recall the Reidemeister II chain maps. Here and in the following, we borrow notation from Soergel bimodules for the webs that appear: Consider the cube of resolutions chain complex for a Reidemeister II tangle: Here, the Koszul signs in the tensor product depend on an (arbitrary) ordering of the crossings of the tangle, which is shown on the left. The Reidemeister II chain maps, which connect the complex R ⊗ R of the trivial tangle to this complex (and vice versa), are given by identities on R ⊗ R, as well as the negative of the following more complicated composite foam (and its reflection in a horizontal plane): An analogous argument applies in the case of two colored circles-it uses an explicit description of the chain maps associated to colored Reidemeister II moves-and shows that the braiding of such is given by the rotation foam generated by the linear combination of webs shown in the proof of Lemma 4.17.
In fact, we expect that the annular Khovanov-Rozansky functors are braided on the entire annular link category, but we do not know how to prove this without assuming a stronger functoriality property, which has not been established yet.  Proof. For k = 2, the anti-symmetrizer is (id − σ)/2. Note that this is exactly 1/2 times the foam shown on the left-hand side of (22). Cutting this foam in half by a horizontal plane produces a merge foam M and a splitter foam S. We have (id − σ)/2 = S • M/2 and M/2 • S = id 2 . This implies that S and M/2 represent the desired mutually inverse isomorphisms in Kar(AFoam + ).
The case k > 2 follows, since the anti-symmetrizers in [S k ] and also the projections onto k-colored essential circles in AFoam + can be constructed from the k = 2 cases in the same way.
Everything in this subsection works in the finite-rank case, i.e. for annular gl N foam categories N AFoam + . In this setting, essential circles of label N + 1 are isomorphic to the zero object, which implies that the uncolored essential circle is of rank (at most) N in the sense of Definition 3.4.
Remark 4.26. An analogue of Lemma 4.25 shows that the framed Khovanov-Rozansky functors − fr send the symmetrizer in [S k ] to the k-colored unknot. This is at odds with our interpretation of that colored circle as corresponding to the exterior power k of the uncolored circle. The origin for this discrepancy is the relative homological shift between the two conventions for crossings (16) and (17), which translates into a sign-twist on the braiding.

Evaluation.
Here we recall the evaluation of annular homology developed by Queffelec and Rose. Let L be an annular link, then Khovanov-Rozansky functor sends it to a complex of webs, and by Corollary 4.20 we can replace it by a complex of Schur functors of E in K b (P). The object E appears as the invariant of the essential planar unknot in the annulus and the endomorphism X encodes information about the [X]-actions in link homologies that are typically associated with the choice of a base point on the link. Proposition 3.6 now immediately implies the following: Theorem 4.27. Let C be an arbitrary additive symmetric monoidal category, and let K b (C) be the corresponding homotopy category. Suppose that E is an object of C with an endomorphism X. Then there is a unique functor AKhR(E, X) : ALink + → K b (C) which factors through the Khovanov-Rozansky functor, sends the essential planar unknot to E, the base point action to X, and the braiding of two unknots to the symmetry on E ⊗ E.
The results of [QR18] can be then rephrased in the following way: Theorem 4.28 ( [QR18]). If C = gr Vect, the category of -graded vector spaces (with the swap symmetry), E = [X]/X N , and X is the endomorphism given by multiplication by x, then the functor AKhR(E, X) agrees with the gl N Khovanov-Rozansky homology. If E = [X]/P (X) for a degree N monic polynomial, then AKhR(E, X) agrees with the deformed Khovanov-Rozansky homology studied in [Wu12,RW16]. If C = gr Rep(U (gl N )) and E = V = N is the vector representation of U (gl N ), then the functor AKhR(E, 0) agrees with the annular Khovanov-Rozansky homology.

Coxeter braids, categorified
5.1. Positive Coxeter braids. The purpose of this section is to prove the following theorem.
Theorem 5.1. Let now C − n and C + n denote the annular complexes of the (n − 1)-fold negatively and positively stabilized unknots. Then we have: Lemma 5.2. Upon evaluation as in Theorem 4.28, these complexes compute the planar gl N Khovanov-Rozansky homologies of stabilized unknots.
Proof. This follows from Example 3.22.
To start the proof of Theorem 5.1, note that it suffices to prove one of the homotopy equivalences. The other one follows by symmetry. We focus on the positive stabilization and consider the shifted complex C + n := q n−1 C + n [n − 1], which has its terminal chain group in homological and q-degree zero. We also define annular complexes X n,l as in Figure 2. Clearly, X n,0 = C + n .
Proof. The isomorphism for X 1,l is due to a bigon removal. To check the homotopy equivalence for X n,l , we resolve the right-most crossing σ n−1 and simplify as follows.  In the second step we have used the bigon relation, and in the last step the square-switch relation. The degree zero components of the differential between the copies of C + n−1 ⊗ l+1 (E) are identities up to non-zero scalars [Wu12, Direct Sum Decomposition (IV)], and after Gaussian elimination, we obtain the claimed form.
Corollary 5.4. There is a natural map a : C + n−1 ⊗ E → X n,0 = C + n . Corollary 5.5. One can write where [n] q n (E) := n (E) ⊕ q 2 n (E) ⊕ · · · ⊕ q 2n−2 n (E). The differential D consists of the internal , and some differentials out of [n] q n (E), as well as possibly some higher differentials.
Proof. We prove by induction the existence of a more general expression where the differentials are described as above. Indeed, for n = 1 we get X 1,l ∼ = [l + 1] q l+1 (E), and for n > 1 we use the induction hypothesis and Now, for l = 0 we get X n,0 = C + n .
Theorem 5.6. We have Note that Theorem 5.1 follows from Theorem 5.6 by grading shifts and symmetry. We will prove Theorem 5.6 by induction in n using the recursive description (23). To illustrate this, we first consider the examples n = 2, 3.
Example 5.8. For n = 3 the complex (23) has the form The degree zero differentials are organized in three subquotient complexes: and q 4 3 (E). Here, in cancelling, we assume that the shown differentials are non-zero. Then we get . The case where some of the shown degree zero differentials are zero can be excluded, because then, in specialization E = [x]/x N , the homology would be larger than expected, contradicting Lemma 5.2.
Proof of Theorem 5.6. Assume that (24) holds for all k < n. The terms in (23) in homological degree k are assembled from the terms in C + n−j ⊗ j (E) in homological degree k + 1 − j. By the induction hypothesis, the latter is homotopic to q 2k+2−2j S n−k−1,1 k+1−j (E) ⊗ j (E). The differential decreases the homological degree k by one, and acts between C + n−j ⊗ j (E) → C + n−j ′ ⊗ j ′ (E) with j ′ ≤ j. For j ′ < j − 1 we get k − j ′ > k + 1 − j, and such a differential can be ruled out as it would have negative q-degree. Therefore, in this presentation, the only surviving differentials are internal for C + n−j (that is, j ′ = j) of q-degree two, and between the neighbors j ′ = j − 1, of q-degree zero. The example for n = 3 above illustrates this point.
This means that we have explicitly identified all differentials in (23) except for the ones connecting the leftmost copies of n (E) to C + n−j ⊗ j (E). To simplify this complex, we first consider the degree zero differentials. In q-degree 2t we get the following complex (ignoring the leftmost term), terminating in homological degree t: The differentials are induced by compositions and, in particular, they are non-trivial. One can check that these cancel almost everything, except for a copy of q 2t n (E) in homological degree (n − 2) and q 2t S n−t,1 t (E) in homological degree t.
We claim that the leftmost term [n] q n (E) cancels all q 2j n (E) except for q 2n−2 n (E). Moreover, the remaining differentials are all non-zero and, thus, determined up to scalars. Both claims hold because otherwise, in specialization E = [X]/X N , the homology would be larger than expected, contradicting Lemma 5.2.
Remark 5.9. The annular sl 2 homology of the stabilized unknot was computed by Grigsby, Licata and Wehrli [GLW18]. It agrees with our computation up to conventions, as we shall now explain. Let V n denote the (n + 1)-dimensional irreducible representation of sl 2 . Note that S n (V 1 ) ∼ = V n , S n−1,1 (V 1 ) ∼ = V n−2 and S n−i,1 i (V 2 ) = 0 for i > 1. Thus, for E = V 1 and x = 0 we evaluate As in [GLW18, Section 9.2], the annular sl 2 homology of the n-fold stabilized unknot consists of the two irreducible sl 2 -representations V n+1 and V n−1 in adjacent homological degrees, with a difference in q-degrees of 2.

Morphisms.
We shall now describe the hom spaces between the annular complexes associated to closures of Coxeter braids. We start by describing basic chain maps between complexes associated to braids. In doing so, we again borrow notation from the theory of Soergel bimodules.
Definition 5.10. Consider the following chain map: is called the skein triangle. Here we have Note that the annular closure of Cone(f ) is precisely Cube 2 . We have seen that Cube 2 ∼ = Cube S2 2 ⊕ Cube sign In other words, under annular closure, the skein triangle splits: Here we want to emphasize that the dashed maps only appear in the annular closure.
Remark 5.11. There are also non-zero cobordism-induced maps R → q σ fr [1] and q −1 σ −1 fr [−1] → R, which can be interpreted as gluing in a twisted band that increases the writhe: In the gl N -evaluations, there also exist non-trivial maps R → q 1−2N σ −1 fr [−1] and q 2N −1 σ fr [1] → R associated to twisted bands that decrease the writhe.
In both cases, these cobordism-induced chain maps are unrelated to the maps in the skein triangle.
Remark 5.12. Under gl N -evaluation, we have a partially topological description of the chain map f in the skein triangle. We start with σ fr and follow the Reidemeister II chain map to σ −1 σσ fr and then a saddle cobordism map to q 1−N σ −1 #H fr where H denotes a positive Hopf link and the connect sum is taken on the new over-strand. Finally, the projection to the top degree generator of the reduced Hopf link homology (which is not cobordism-induced) induces an onward map to σ −1 fr . The composition is the chain map f .
Proof. Clearly, Hom Sn ( k U n , C[x 1 , . . . , x n ]) is free over x i , so we can consider Hom Sn ( k U n , C[U n ]) instead. It can be identified with the space of S n -invariant differential forms on U n , which by a theorem of Solomon [Sol63] is isomorphic to the space of differential forms on U n /S n = Spec [p 2 , . . . , p n ].
Let Cube n , as before, denote the Koszul complex for x i − x i+1 acting on E ⊗n . Then we have: Theorem 5.14. The morphisms between the tensor products of Cube n can be described as follows: • The endomorphism algebra of Cube n is: where U n is the (n − 1)-dimensional reflection representation of S n and S n acts trivially on x.
Moreover, x is of q-degree 2 and U n is supported in homological degree −1 and q-degree −2. • The spaces of morphisms between Cube n ⊗ Cube m and Cube n+m are generated by the canonical maps implicit in the description of . The actions of x ∈ End(Cube i ) on this space agree (up to homotopy) for i = n, m, n + m, and the actions of exterior algebras are naturally identified under the induction and restriction maps between U n ⊕ U m and U n+m .
• All other morphisms are induced by these.
Proof. Let us compute the endomorphism ring of Cube n . We have End(E ⊗n ) = [x 1 , . . . , x n ] ⋊ [S n ], so End(Cube n ) is isomorphic to a complex built out of these. Since the differential does not involve the action of S n , we can ignore the [S n ] factor for a while. Now Here we have identified U n with its dual via the S n -invariant nondegenerate bilinear form. This space of maps carries the natural differential where d is the Koszul differential on • (U n ) ⊗ [x 1 , . . . , x n ]. Since (x 1 − x 2 , . . . , x n−1 − x n ) is a regular sequence in R = [x 1 , . . . , x n ], the homology of ( • (U n ) ⊗ R, d) is isomorphic to [x]. Equation (25) presents D as a sum of two anticommuting differentials, which induces a spectral sequence. The first differential has homology • (U n ) ⊗ [x] ⋊ [S n ]. Now the second differential vanishes, so the spectral sequence collapses at E 2 page, and Here x has q-degree 2 and homological degree 0 while the generators ǫ 1 , . . . , ǫ n−1 of • (U n ) have homological degree −1 and q-degree −2. See also Example 5.17 for an alternative computation of End(Cube n ).
We can apply the same method in a more general situation. To compute Hom(⊗ i Cube ni , ⊗ j Cube mj ) (with n i = m j = n), we first observe that both complexes consist of several copies of E ⊗n . Now we replace the Hom space between two such copies by End(E ⊗n ) = [x 1 , . . . , x n ] ⋊ [S n ], and write two sets of differentials. The first differential is given by multiplication by x i − x i+1 if i, i + 1 are in the same block of the partition n = n i , and the second is given by multiplication by x i − x i+1 if i, i + 1 are in the same block of the partition n = m i .
Although the differentials do not involve [S n ], we still need to keep track of its action. Let denote the space of polynomial maps, that is, the ones induced by the polynomial action on E ⊗n . Any endomorphism of E ⊗n can be uniquely written as f = σ∈Sn f σ σ for some polynomials f σ . Similarly, any morphism Hom(⊗ i Cube ni , ⊗ j Cube mj ) can be uniquely written as f = σ∈Sn f σ σ where f σ are polynomial chain maps. The space of such f σ is isomorphic to Hom pol (σ(⊗ i Cube ni ), ⊗ j Cube mj ). To sum up, to describe all morphisms between products of cubes it is sufficient to describe the polynomial morphisms between products of cubes where the variables in one product are possibly relabeled. Note that before we did not have this problem since S n preserves Cube n .
After relabeling, we get two set partitions Π and Π ′ with r blocks of size n i and s blocks of size m j respectively. We will refer to the products of cubes as to Cube Π and Cube Π ′ . Let Π ′′ be the finest set partition which is a coarsening of both Π and Π ′ . If Π ′′ has more than one block then Hom pol (Cube Π , Cube Π ′ ) factors over the blocks of Π ′′ and we can proceed by induction.
From now on we will assume that Π ′′ = {1, . . . , n}. Let us compute Hom pol (Cube Π , Cube Π ′ ) using the spectral sequence as above. After applying the first differential we get a polynomial algebra with one variable per block in Π. After applying the second differential we identify all these variables and obtain an exterior algebra with generators ǫ ij for all i, j such that i, j are in the same block in both partitions Π, Π ′ .
We can describe all these chain maps and their gradings more explicitly. Recall that so there are natural chain maps By combining these, we get maps Every polynomial morphism from Cube Π to Cube Π ′ can be obtained as a composition of these merge and split maps with a polynomial endomorphism in End pol (Cube n ) = • (U n ) ⊗ [x]. In particular, the identity on Cube n induces a chain map of q-degree s + r − 2 and homological degree s − 1. The odd variable ǫ ij can be identified with ǫ i + . . . + ǫ j−1 in U n dual to which acts on Cube n . Note that if i and j are not in the same block for Π or Π ′ then ǫ ij acts by 0. For i, j, k in the same block for both partitions Π, Π ′ the actions of ǫ ij , ǫ jk and ǫ ik satisfy an obvious linear relation.
Example 5.16. Let us describe the endomorphisms of In homological degree zero we have [x 1 , x 2 ] ⋉ [S 2 ]. In homological degree −1 we have a chain map ǫ of q-degree −2 which sends the first copy of E ⊗2 to the second one, and the right copy to zero: Note that in this case the projection to the first copy of E ⊗2 yields the split map Cube 2 → qE 2 [1] while the inclusion of the second copy yields the merge q −1 E 2 → Cube 2 . The composition of split and merge coincides with ǫ. There is also another map h of homological degree one: So the endomorphism ring of Cube 2 in the homotopy category is isomorphic to Example 5.17. Similarly to Example 5.16, for Cube n we have chain maps ǫ 1 , . . . , ǫ n−1 and homotopies h 1 , . . . , h n−1 , and [d, (ǫ 1 , . . . , ǫ n−1 ) ⋊ [S n ]. As above, ǫ i span a copy of the reflection representation U n .
Example 5.18. Let us illustrate the difference between polynomial morphisms (which were discussed in the proof of Theorem 5.14) and all morphisms. For example, let us compute Hom pol (Cube 1 ⊗ Cube 2 , Cube 2 ⊗ Cube 1 ).
We have the following diagram: Here ǫ has q-degree −2 and homological degree −1.
Alternatively, ǫ can be obtained as a composition of the split and merge maps: Finally, observe that the transposition (1 3) ∈ S 3 yields an obvious degree zero isomorphism between Cube 1 ⊗ Cube 2 and Cube 2 ⊗ Cube 1 . This isomorphism is not polynomial, in fact, the above computation shows that there are no polynomial morphisms of degree zero.

By construction, [d,
Remark 5.21. It is easy to see that the split and merge maps between Cube n induce similar split and merge maps between C + n . Since ǫ i can be obtained as a composition Cube n → qCube i ⊗ Cube n−i [1] → q 2 Cube n [1], the seemingly mysterious endomorphisms ξ k can be obtained as sums over all i of compositions for some explicit polynomials φ i,k (x) of degree k.
It is likely that all morphisms between various tensor products of C + n are generated by splits, merges and the action of polynomials. It would be interesting to describe all relations between these morphisms, categorifying Turaev's description of the skein of the annulus (Theorem 2.1). We plan to pursue this in a future work.
Example 5.22. Let us describe the maps from Since C − 2 is a summand in Cube 2 , every such map factors through Cube 2 . Thus we have a map Let e λ ∈ S n denote our chosen Young symmetrizer in [S n ] of shape λ. As before, we denote by Cube λ n = e λ Cube n the direct summand of Cube n cut out by the action of e λ . Then we have the following corollary of Theorem 5.14.
Remark 5.23. The category K b (P) has a t-structure, whose heart is given by the complexes whose chain groups are q-shifted by twice the homological degree. Cube n as well as all Cube λ n are shifted perverse. 5.3. Other Coxeter braids. To categorify the formula from Theorem 2.20, we would like to give a more categorical perspective on ribbon skew Schur functions, following Solomon [Sol68]. Given a binary sequence ǫ of length n, we can define two parabolic subgroups W ǫ , W ′ ǫ of S n generated by simple reflections with positive (resp. negative) signs. Let s ǫ and s ǫ denote the symmetrizer for W ǫ and antisymmetrizer for W ′ ǫ . Theorem 5.24 ( [Sol68]). The group algebra [S n ] can be presented as a direct sum of left ideals: Furthermore, the character of the S n -representation [S n ]s ǫ s ǫ equals the ribbon skew Schur function Ψ(a) for the composition a corresponding to ǫ.
We denote by p ǫ ∈ [S n ] the idempotent projecting to [S n ]s ǫ s ǫ . Now we are ready to describe the annular invariants of the Coxeter braids σ ǫ = σ ǫ1 1 · · · σ ǫn−1 n−1 . Theorem 5.25. The annular complex C ǫ of the Coxeter braid σ ǫ is determined by Proof. We induct on the length of ǫ and the number of minus signs in ǫ. Suppose there is just one minus sign in the a-th place. Then the skein triangle gives us a homotopy equivalence: ] Now we use that the right-hand side is a direct summand in . We also know that C + n [n − 1] is a direct summand in Cube n cut out by the idempotent p n ∈ [S n−1 ]. The projection onto this summand Cube n → C + n [n − 1] factors through the right hand side of (27), so the skein triangle (27) splits and C + n [n − 1] is a direct summand in the right hand side. Hence C ǫ [n − 2] is also a direct summand in Cube n defined by the difference of the two idempotents p n − p a ⊗ p b = p ǫ .
A similar argument works for the induction step. Here we use the skein triangle to get: Then we use the induction hypothesis to find C α ⊗ C β [|ǫ| + ] as a direct summand in Cube a ⊗ Cube b , such that the inclusion intertwines the operators x−y. If follows that the cones are direct summands of Cube n , and so is C ǫ ′ [|ǫ ′ | + ] and hence C ǫ [|ǫ| + ].
It remains to check that the projectors for all these summands agree with p ǫ . Indeed, they can be computed recursively by successively subtracting induced smaller projectors from the bigger ones (this categorifies (4)). On the other hand, p ǫ satisfy the same recursion by [Sol68, Theorem 3].
We are now in a position to prove a conjecture of Hunt-Keese-Licata-Morrison about the annular Khovanov homology of Coxeter braids and the spectral sequence to planar Khovanov homology.
For this, we will use that the annular Khovanov homology can be computed via annular evaluation along the functor AKhR(V 1 , 0) where V 1 is considered as the vector representation of sl 2 with graded dimension qz + q −1 z −1 and z encodes the weight space grading. The planar Khovanov homology can similarly be obtained via AKhR(V 1 , e), with e ∈ End(V 1 ) provided by the sl 2 action. The spectral sequence from annular to planar Khovanov homology arises by filtering AKhR(V 1 , e) along the weight space grading.

Annuli in tangle diagrams
In this section we study applications of annular evaluation to Khovanov-Rozansky invariants of tangles which contain a cabling of a framed unknot as a sublink. This includes tangles obtained by wrapping an annular link around a tangle as in (5).
6.1. A symmetric group action on cables. The following theorem is due to Grigsby-Wehrli-Licata in the context of Khovanov homology [GLW18]. The version here applies to all sufficiently functorial Khovanov-Rozansky link homologies of type A.
Theorem 6.1. Let T be a link or a tangle, which has n parallel closed 1-colored components. Then T carries an action of Br n by endo-cobordisms that braid these parallel components around each other. Let KhR denote a Khovanov-Rozansky-type invariant, which is functorial under such cobordisms 4 . Then the induced action of Br n on KhR(T ) factors through S n .
Proof. It suffices to prove that the braiding is symmetric on two parallel components. We have already seen this in the proof of Theorem 4.23 for the case when L has the two components as a disjoint split factor. Now, we consider the general case, which can be modelled as follows. (28) Here, T is compressed into the small box shown, except for the two parallel components in question (if the tangle is not a link, then some additional strands might connect this box to the boundary). The braiding of the two circles starts with a Reidemeister II move, followed by isotoping the 1-labeled crossing all the way through the rest of T , and then eliminating both crossings again by an inverse Reidemeister II move. Contrary to the case treated in Theorem 4.23, we do not intend to compute this chain map σ explicitly. We only need to show that it is equivalent to its inverse σ −1 , which has a movie description as in (28), except that the bottom strand passes over the top strand first, and it is a negative crossing instead of a positive crossing that slides all the way through T . In the absence of other components, we have seen that these chain maps are plainly equal, since the different Reidemeister II chain maps uses in these variants agree (up to to cancelling signs).
In the present case, we additionally have to take into account moving the crossing labeled 1 through the box, i.e. through the rest of T . There are three key observations which allow to compare the contributions of this process to σ and σ −1 .
First, isotoping the crossing through the rest of T is realized as a sequence of braid-like Reidemeister III moves. A braid-like Reidemeister III move is one in which the relevant local tangle 6-ended tangle has the following sequence of boundary orientations up to cyclic reordering: out-out-out-in-in-in. In contrast, a star-like Reidemeister III tangle would have an alternating sequence of boundary orientations out-in-out-in-out-in.
Second, the intermediate chain complexes in (28) can be seen as total complexes of double complexes, with a horizontal differential contributed by the crossings in T , and a vertical differential contributed by the extra crossings created by the initial Reidemeister II move. Note that the initial and the final chain complex in this sequence are supported in the single vertical degree zero.
Third, the chain maps associated to the braid-like Reidemeister III moves are filtered with respect to the vertical degree. This means that these chain maps are sums of components that preserve the vertical degree, and components which, at most, increase the vertical degree, but never decrease it. Moreover, in a pair of Reidemeister III moves, which differ only in the sign of the 1-labeled crossing which is pushed under (or over) another strand in T , the filtration-preserving components agree. For 1-colored strands, this is well-known to experts and can be read off from the explicit descriptions of Reidemeister III chain maps for Rouquier complexes in [EK10]. The general case follows via the strategy of exploding strands of higher color into 1-colored strands before sliding the crossing, see e.g. [Wu14, Section 14.1].
The chain maps obtained by isotoping a positive or a negative crossing through the rest of T are both filtered, and their filtration-preserving components agree. Finally, σ and σ −1 are obtained from these chain maps by pre-and postcomposing with Reidemeister II chain maps. Since the latter have non-zero components only in vertical degree zero, these composite only depend on the filtration-preserving parts of the intermediate Reidemeister III chain maps. As noted above, these agree.
In particular, Theorem 6.1 holds for the the gl N Khovanov-Rozansky invariants T and T fr valued In the following, we write T = T (E ⊗n ) for tangles as in the theorem.
Proof. By Theorem A.10, we have that K b (Kar(N Foam)) is Karoubian. By Theorem 4.7 there is a braid group action on T (E ⊗n ) fr , which factors through the symmetric group. Hence by Proposition A.12 the Schur functors are well defined up to homotopy equivalence. Proof. The [S n ] part is obtained by linearising the symmetric group action from the theorem. In the polynomial part, x i acts by a dot on the i-th component of the cable. The proof of the theorem and the fact that dots slide through crossings up to homotopy implies that stated compatibility.
We can summarize the two corollaries as follows.
Corollary 6.4. Each tangle as in Theorem 6.1 provides an additive functor fromP to K b (Kar(N Foam)).
As before, we also get a version of Lemma 4.25 in the presence of other strands. For this, let T (E n ) denote the tangle T with a n-colored component in place of the n parallel uncolored components.
Proof. The proof proceeds analogous to the one for Lemma 4.25 by identifying the chain map for the k = 2 anti-symmetrizer on T (E ⊗ E) fr with the projection onto T (E 2 ) fr .
This implies that cobordism-induced braiding is also symmetric for colored circles, as proved for uncolored circles in Theorem 6.1.
6.2. Annular simplification. If an annular link L appears as a sublink of a tangle T which is a cabling of a framed unknot, then the associated Khovanov-Rozansky chain complex T fr can be simplified to a complex in which the annular link L is replaced by the a complex of -colored concentric circles or Schur functors. Here we prove that this induces filtrations and spectral sequences as claimed in Theorem 1.8 and Corollary 1.9.
Proposition 6.6. Let L denote a link diagram in the thickened annulus, T a tangle diagram with a blackboard-framed unknot component without self-crossings, and T (L) the tangle diagram obtained by cabling this unknot component in T by L. Then the chain complex T (L) fr is isomorphic in K b (N Foam) to a filtered chain complexC, whose associated graded is isomorphic to a formal direct sum of grading shifts of chain complexes of the form T (CC) fr where CC denotes the collections of concentric -colored circles that appear in the annular simplification of L. Moreover, the component of the differential that increases the filtration degree by one is induced by the corresponding annular differential.
Interesting examples of tangles T (L) are tangles obtained by wrapping as in (5) and cabled Hopf links H(L, L 2 ) as in the introduction.
Proof. We write C := T (L) fr . The key idea of the proof is that annular simplification is still possible in the presence of additional strands. Indeed, the annular simplification algorithm of Queffelec-Rose [QR18, Proposition 5.1] utilizes two types of web isomorphisms, which both continue to hold in these settings: namely certain local isomorphisms (rung combination and square switch) which hold on the nose, and the global rung slide move, which uses fork-slide moves in the presence of additional strands.
The chain complex C can be viewed as a total complex of a tricomplex with one direction (horizontal) corresponding to crossings internal to the annular link L 1 , the second direction (vertical) to crossings of that annular link and the rest, and the third direction (depth) to crossings purely in the rest. Since the third direction will not play an important role, we will suppress it and consider C as total complex of a bicomplex C * , * . The columns C * ,i in such bicomplexes are complexes in their own right, which are isomorphic to the invariants of the annular webs appearing in the cube of resolutions of L 1 , interacting with other additional link and tangle components.
By annular simplification, each column C * ,i is homotopy equivalent to the total complexC * ,i of a bicomplex whose columns are of the form T (CC), where CC is a collection of concentric circles. Now we substitute the columns in the bicomplex C * , * by the homotopy equivalent complexesC * ,i . In doing so, we collapse the two "horizontal" directions: the one already present in C * , * and the additional direction in eachC * ,i . Because of the column substitutions,C * , * will typically no longer be a bicomplex. Besides the vertical differential d 0 :C * , * →C * +1, * and the horizontal component d 1 :C * , * →C * , * +1 , there are now also higher components d k :C * , * →C * +1−k, * +k . In Figure 3, we illustrate the result of a single column substitution.
The perturbed bicomplexC * , * still carries the horizontal filtration F j = j ′ ≥jC * ,j ′ , whose associated graded is isomorphic to the direct sum of the columns, with differential d 0 , which we identify with the invariants of collections of colored circles interacting with the remaining strands. The filtration degree one component of the total differential is d 1 and its components originate from crossings in the annular link or the resolution of annular webs by concentric circles-in this sense, it is induced by the annular differential computed by the annular simplification algorithm of Queffelec-Rose.
Example 6.7. If we apply Proposition 6.6 in the case of the Hopf link cable H(L 1 , ∅), we obtainC * , * = C 0, * and the only non-trivial component of the differential is d 1 .
Corollary 6.8. The complexC from Proposition 6.6, considered as an object of K b (Kar(N Foam)), can be decomposed further into a filtered complex C ′ with associated graded given by Schur-colored unknots interacting with the remaining strands.
Proof. This follows from Proposition 6.6 and Corollary 6.5, which identifies colored circles with tensor products of antisymmetric Schur functors, which we can then decompose further.
This completes the proof of Theorem 1.8 and implies Corollary 1.9.
Remark 6.9. Corollary 6.8 can also be proved directly following the strategy of the proof of Proposition 6.6, but with the Queffelec-Rose annular evaluation algorithm replaced by the alternative annular evaluation algorithm outlined in Remark 4.21.
6.3. Generalized Hopf links, categorified. Here we show how the above results can be used to compute Khovanov-Rozansky homologies of generalized Hopf links. First, we consider annular links wrapped around a single vertical strand colored by k . We reduce on this vertical strand, so that the corresponding tangle has no non-trivial endomorphisms, and the invariants in question are valued in complexes of graded vector spaces. For the definition of reduced colored Khovanov-Rozansky homologies, we refer to [Wed19].
Theorem 6.10. Let L be an annular link diagram and let T ( i , L) be the tangle consisting of L wrapped around the reduced vertical strand colored by i . Consider the following bigraded vector space with an action of [X]: Then there is a spectral sequence with the E 2 page given by the evaluation of the annular complex of L at E ∧ i and E ∞ page isomorphic to T ( i , L) fr .
If L is a single j -colored unknot, then the invariant of T ( i , L) was computed by second author in [Wed16,Proposition 4.15]. For j = 1, it agrees with E ∧ i as a bigraded vector space. The action of the dot on L can be easily computed, and it agrees with the action of X above.
Suppose that now L is an arbitrary annular link. By Proposition 6.6 the Khovanov-Rozansky complex of T ( i , L) is filtered with associated graded given by the evaluation of the annular complex of L at (E ∧ i , X). More precisely, the differential splits into two parts: the annular differential d ann for L and the additional differential d wrap responsible for the crossings between the webs in the resolution of L and the vertical strand. We get a spectral sequence by first applying d wrap and then the induced differential d * ann . It converges to the homology of the total complex. By Proposition 6.6 the homology with respect to d wrap is isomorphic to the evaluation of the annular complex for L at (E ∧ i , X) (with no differential). On the next page of the spectral sequence we compute the homology with respect to d * ann , which is just the homology of the annular complex for L evaluated at (E ∧ i , X).
Corollary 6.11. Let L be a i -colored unknot, and T ( i , j ) = T ( i , L) as above. Then the Khovanov-Rozansky homology of T ( i , j ) is isomorphic to j (E ∧ i ) as above. In particular, its graded dimension is This agrees with [Wed16, Proposition 4.15].
We expect that Theorem 6.10 can be generalized to other projectors, categorifying Lemma 2.25. Specifically, Elias and Hogancamp recently constructed [EH17] a family of projectors P λ in the homotopy category of Soergel bimodules which categorify the projectors p λ from Section 2. These are idempotent complexes which are bounded from above. Let P λ be the smallest triangulated subcategory of the homotopy category containing P λ . After specialising to the gl N theory, we expect the following.
Conjecture 6.12. Let L be an annular link and let T (P λ , L) denote the tangle consisting of L wrapped around P λ . Then T (P λ , L) fr is an object of the category P λ . If L is a single unknot then for some differential D.
Example 6.13. If λ = (1 i ) then (29) can be interpreted as saying that L acts on P λ with "eigenvalue" This agrees with Theorem 6.10.
Remark 6.15. This conjecture gives a precise categorical context to the "refined S-matrix" defined by Aganagic and Shakirov, see [AS15]. Specifically, they conjecture that (a) the projectors P λ in certain sense correspond to Macdonald polynomials H λ (x; q, t), and (b) the "refined Chern-Simons invariant" of the generalized Hopf link with components labeled by P λ and P µ equals While we are unable to comment on (a) at the moment, we can interpret (b) by cutting the component with P λ open. Then the invariant of the corresponding tangle equals Since this is linear in H µ , we can instead consider a tangle where one component is colored by P λ and the other is a closed circle colored by an arbitrary symmetric function f . The "refined Chern-Simons invariant" of this tangle equals which agrees with a certain decategorification of (29).
We would like to comment on possible (but yet mostly conjectural) connections between the results of this paper and the work of the first author, Negut , and Rasmussen [GNR21], as well as the series of papers of Oblomkov and Rozansky [OR18, OR19, OR20]. One of the main conjectures of [GNR21] assumes the existence of a monoidal functor where Hilb n ( 2 ) is the Hilbert scheme of n points on the plane and D b Coh denotes the derived category of coherent sheaves. On the Hilbert scheme of points we have two important sheaves: T is the tautological bundle of rank n while I is the tautological ideal sheaf (of infinite rank). The fibers of T and I over a given ideal I ⊂ [x, y] are equal to [x, y]/I and to I, respectively. Both T and I enjoy the action of two commuting endomorphisms X and Y .
Conjecture 6.16. The gl N invariant of a single unknot wrapped around n vertical strands is isomorphic to the gl N reduction of the object ι * (I/(Y, X N )I).
As explained in [GNR21], the projectors P λ should correspond to the fixed points of the torus action on Hilb n ( 2 ), that is, to the monomial ideals I λ t 5 . At such a monomial ideal, the fiber of I/(Y, X N )I has a bigraded character which agrees with (29). This means that Conjectures 6.12 and 6.16 are compatible with each other. See also [Nak14] for more detailed relation between the refined S-matrix and the geometry of the Hilbert scheme of points.
Finally, we would like to comment on the relation between this work and [Eli18]. There, Elias constructed a family of objects X λ (labeled by Young diagrams λ) in the Drinfeld center of the category of (extended) affine Soergel bimodules. It is expected that X λ descend to the homotopy category of Soergel bimodules, and their images are filtered by the products of Jucys-Murphy braids L i according to the weight decomposition of the irreducible representation V λ of gl N . For example, for λ = the complex X is filtered by L i , each with multiplicity one.
We expect X λ to be closely related, but not identical to our annular links wrapped around vertical strands. In the notations of Conjecture 6.16 we expect X λ = ι * (S λ (T )), in particular, X = ι * (T ). This relation is expected to categorify Lemma 6. 6.4. A note on wrapping. The initial motivation for this paper was to categorify the wrapping operation (1). In HOMFLY-PT skein theory, the action of encircling braids by positive annular links descends to an action of the cocenter of all Hecke algebras H m of type A on the center of H n ⊗ . On the topological level, and with a view towards categorification, the encircling operation can be described as a functor from ALink + , the 1-cocenter (horizontal trace, see Section 7.1) of the braided monoidal 2-category of braids and their cobordisms, to the centralizer Z(Braid n ) of the 2-category of braids (and their cobordisms) on n coherently oriented strands inside the 2-category of tangles Tan n with the same boundary data.
In this paper, we have described and studied the universal target for the currently available Khovanov-Rozansky functors for positive annular links, namely the category K b (P). The categorified analog of the Hecke algebra H n is the homotopy category of Soergel bimodules K b (SBim n ). A first approximation to what a categorification of the wrapping operation could be is given in Figure 4, ignoring the second column. Unfortunately, K b (P) does not seem to be rich enough to admit a functor to the Drinfeld center Z(K b (SBim n )) that intertwines the Khovanov-Rozansky functors for annular links and partial braid closures, as we will explain next.
Example 6.17. Let L be an annular link and T a tangle. Consider the cobordism that rotates L once around the annulus. This cobordism induces the identity map on the annular invariant in K b (P). However, after wrapping L around the tangle T , the cobordism that rotates L around T is not expected to induce the identity map on the tangle invariant in K b (SBim n ).
To get a categorified wrapping operation, we thus need an upgraded annular Khovanov-Rozansky functor with a target category that remembers such rotation cobordisms. A natural candidate for such a category is the derived horizontal trace of the dg category SBim dg n of complexes of Soergel bimodules. This and a related notion of derived center feature in the second column of Figure 4 and are the focus of the follow-up paper [GHW21] of the authors with Matthew Hogancamp. 7. Traces outside of type A 7.1. Categorical traces. We briefly review the definitions of categorical traces following [BHLŽ17].
If C is a k-linear category, its vertical trace vTr(C) (also known as zeroth Hochschild homology) is a k-vector space spanned by all possible f ∈ End C (X) for X ∈ Ob C modulo the relations f g ∼ gf for any f ∈ Hom(X, Y ) and g ∈ Hom(Y, X). If C is monoidal then vTr(C) has a natural algebra structure.
If C is a monoidal k-linear category, one can also define its horizontal trace hTr(C). This is a k-linear category where the objects are the same as the objects in C, and the morphisms are defined by where for any f ∈ Hom C (X ⊗ Z, W ⊗ Y ) and g ∈ Hom C (W, Z) we identify the compositions

It is easy to see from this definition that
Hom hTr(C) (1, 1) = vTr(C).
For the definition of composition of morphisms and further details we refer to [BHLŽ17]. There is a natural trace functor hTr : C → hTr(C) which sends any object of C to the namesake object in hTr(C). If C has duals then hTr(X ⊗ Y ) ≃ hTr(Y ⊗ X).
Informally, one can think of objects of hTr(C) as of annular closures of objects in C. In particular, the horizontal trace for the category of webs (with morphisms given by foams) is the category of annular webs (with morphisms given by annular foams).
Finally, there is a derived version of the above definitions developed in detail in the follow-up paper [GHW21]. The vertical trace is replaced by full Hochschild homology of C, while the horizontal trace becomes a dg category.
k≥0 Sym k (h * ⊗ ) denote the coordinate ring of the representation, i.e. the polynomial ring generated by the simple roots. We let SBim (and SSBim) denote the category of (singular) Soergel bimodules associated to (W, S) and the above realization.
Definition 7.1. Let Cube W denote the Koszul complex of [W ]-modules determined by its degree one differential h * ⊗ R → R given by multiplication m : α s ⊗ x → α s x. More explicitly with differentials induced by co-multiplication and multiplication Recall that R is the monoidal unit in SBim, and we will think of Cube W as a complex in K b (SBim), and in particular, as a complex of R − R-bimodules. To this, we can apply the horizontal trace functor term-wise. Here, the horizontal trace is nothing but HH 0 , i.e. the functor of tensoring with R over R ⊗ R, which identifies the left-and right actions. We consider the resulting objects as R-modules.
Since Cube W is built from copies of R and hTr(R) = R, we could identify it with its image hTr(Cube W ) in K b (hTr(SBim)). Note that [W ] ⋉ R acts on R and the differentials in hTr(Cube W ) are equivariant for this action, so we will consider hTr(Cube W ) as a complex of [W ] ⋉ R-modules.
Definition 7.2. Let C ǫ denote the chain complex in K b (hTr(SBim)) obtained from the (suitably normalized 6 ) Rouquier complex [Rou06] of σ ǫ by applying the horizontal trace functor term-wise.
Let ǫ ∈ {±1} k and partition the set S = S + ⊔ S − according to the chosen order of simple roots. We denote the corresponding parabolic subgroups by W ǫ and W ′ ǫ . Let s ǫ ∈ [W ] be the symmetrizer corresponding to W ǫ , and s ǫ ∈ [W ] the anti-symmetrizer corresponding to W ′ ǫ . The following is a generalization of Theorem 5.24. In particular, we expect that C −1,··· ,−1 is homotopy equivalent to hTr(Cube W ) sign and C +1,··· ,+1 [r] is homotopy equivalent to hTr(Cube W ) W . 7.3. Annular simplification in other types. In type A n−1 , we know (and have made ample use of the fact) that Kar(hTr(SBim)) ∼ = [S n ] ⋉ R − gpmod. In this section, we pursue an analogous description for other finite Coxeter groups.
A key tool is Elias-Lauda's computation [EL16] of the vertical trace decategorification of SBim. To describe this, we consider SBim * , the category whose objects are objects in SBim without grading shifts, and hom spaces are graded by Hom SBim * (A, B) ∼ = m Hom SBim (A, q −m B). The vertical trace is the quotient where the span is taken over pairs of f ∈ Hom SBim * (A, B) and g ∈ Hom SBim * (B, A). Since SBim * is graded and monoidal, vTr(SBim * ) has the structure of a graded algebra. Recall that the 2-category of singular Soergel bimodules SSBim for (W, S) is the closure under grading shifts, taking direct sums and summands of the 2-category of bimodules generated by singular Bott-Samelson bimodules R I ⊗ R I∪J R J for I, J ⊂ S. Here we denote by R I the ring of invariants for the parabolic subgroup W I ⊂ W generated by reflections in I.
We identify the objects of SSBim with subsets I ⊂ S and 1-morphisms from J to I are R I − R Jbimodules. The full 2-subcategory of SSBim generated by the object ∅ ⊂ S is canonically identified with SBim. We can think of SSBim as a partial idempotent completion of SBim in the 1-morphism direction.
Lemma 7.6. Considering the vertical trace of the bicategory SSBim as an idempotented algebra, we have an algebra isomorphism given on idempotent truncations by ψ : vTr(SSBim(I, J)) ∼ = 1 I ( [W ] ⋉ R)1 J . Here I and J denote subsets of S and 1 I and 1 J are the corresponding symmetrizers in [W ].
Proof. All singular Soergel (R I , R J )-bimodules B, B ′ can be turned into ordinary Soergel bimodules by tensoring on both sides with R, considered as an (R, R I )-bimodule or as an (R J , R)-bimodule respectively. For I ⊂ S we let r I denote the rank of R I ⊗ R ⊗ R I as a free R I -module (this is the size of the double coset W I \W/W I ). For morphisms B f − → B ′ g − → B, we now define . It is straightforward to check that this defines an algebra map and ψ agrees with φ on the traces of endomorphisms of Soergel bimodules. Furthermore, the image of vTr(SSBim(I, J)) under ψ lands in 1 I ( [q ±1 ][W ] ⋉ R)1 J since φ([id R⊗ R I R ]) = r I 1 I ∈ [W ]. An analogous argument shows that ψ is injective and surjective.
In type A, these functors yield a subcategory of the horizontal trace generated by collections of circles colored by i , and by complexes thereof. Proposition 4.13 then implies that the latter functor is an equivalence. As explained in Section 4, this is related to the fact that every representation of S n can be resolved by representations induced from the trivial representations of the parabolic subgroups.
Outside of type A, this is no longer true. For example, if W = I n is a dihedral group of order 2n, then it has four parabolic subgroups {e}, {s}, {t}, W . The corresponding induced trivial representations have dimensions 2n, n, n and 1 and it is easy to see that for n > 3 the irreducible two-dimensional representation h cannot be resolved by these. On the other hand, by Conjecture 7.4 the horizontal trace of the positive Coxeter braid corresponds to the complex 2 h → h → triv , where we identify an irreducible representation τ of W with Hom W (τ, hTr(R)). Therefore we do not expect the functor K b (vTr(SSBim)) ֒→ K b (hTr(SBim)) to be essentially surjective.
Remark 7.8. The category K b (hTr(SBim)) is expected to be closely related to the category of character sheaves [Lus84,RR21] in the corresponding type. The object E, or the trace of identity object in SBim, corresponds to so-called Springer sheaf, and its endomorphisms match the vertical trace vTr(SBim * ) from Theorem 7.5. In type A, it is known that the summands of the Springer sheaf generate the category of character sheaves, but this is no longer true in other types due to the existence of so-called cuspidal sheaves.
By analogy, we do not expect the functor K b (vTr(SSBim)) ֒→ K b (hTr(SBim)) to become essentially surjective even after Karoubi completion on both sides. The cuspidal sheaves should correspond to certain objects in Kar(K b (hTr(SBim))) which do not belong to the essential image of Kar(K b (vTr(SSBim))). It would be very interesting to construct cuspidal objects explicitly by Soergel-theoretic methods. See also [GHW21, Section 1.4] and [BT22b, BT22a] for a further discussion.
Appendix A. Some facts from homological algebra A.1. Thomason's theorem. Suppose that C is a full triangulated subcategory of a triangulated category A. Following Thomason [Tho97], we say that C is dense in A if every object of A is a direct summand of an object isomorphic to an object in C.
Theorem A.1 ( [Tho97]). Let A be a triangulated category. There is a bijective correspondence between full dense triangulated subcategories of A and the subgroups of the Grothendieck group K 0 (A).
Given a subgroup H ⊂ K 0 (A), the corresponding full subcategory C H consists of objects of A with equivalence classes in H. The theorem states that C H is actually triangulated and dense in A, and all full dense triangulated subcategories appear this way.
Corollary A.2. Suppose that C is a full dense triangulated subcategory of A and K 0 (C) = K 0 (A). Then C = A.
A.2. Strict idempotents. Suppose now that C is Karoubian. Suppose that we are given an idempotent endomorphism ǫ : X → X. Then ǫ = 1 − ǫ is also an idempotent. There is a canonical splitting (31) X = X ǫ ⊕ X ǫ such that ǫ = id on X ǫ and ǫ = 0 on X ǫ .
Lemma A.3. Suppose that a : X → Y is a morphism in C, and X, Y have idempotent endomorphisms ǫ (which we will denote by the same letter) such that aǫ = ǫa. Then a preserves the splitting (31).
Proof. Since C is additive, a morphism between direct sums is determined by its components. It is easy to see that the components X ǫ → Y ǫ and X ǫ → Y ǫ vanish.
Similarly, if A is a chain complex over C and ǫ : A → A is an idempotent chain endomorphism of A then by Lemma A.3 we have a splitting A = A ǫ ⊕ A ǫ . If f : A → B is a chain map and A, B have two idempotent endomorphisms ǫ : A → A and ǫ : B → B then f preserves the splitting.
Remark A.4. Here we need to use that ǫ 2 = id exactly, not up to a homotopy.
Suppose now that C is not only Karoubian, but also symmetric monoidal.
Theorem A.5. Suppose that A and B are two chain complexes over C, then the following are true: (b) If f and g are homotopic then S λ (f ) and S λ (g) are homotopic (c) If A and B are homotopy equivalent, then so are S λ (A) and S λ (B).
Proof. For (a) observe that there is an S n -equivariant morphism f ⊗n : A ⊗n → B ⊗n . Since it is S nequivariant, it commutes with all the idempotents in [S n ], and hence defines a map between Schur functors. For (b), observe that there is an S n -equivariant homotopy between f ⊗n and g ⊗n , so it defines a homotopy between S λ (f ) and S λ (g). Finally, (c) is a straightforward consequence of (b).
A.3. Homotopy idempotents. Recall that to any additive category K, one can associate another category Kar(K) called its Karoubi completion. The objects of Kar(K) are pairs (A, e) where e : A → A is an idempotent. A morphism between (A, e) and (A ′ , e ′ ) is a morphism f : A → A ′ such that f e = e ′ f = f . There is a natural functor i : K → Kar(K) which sends A to (A, id A ).
Proposition A.6. Let K be an additive category, let Kar(K) denote the Karoubi completion of K. Then Kar(K) is additive and Karoubian. The natural functor i : K → Kar(K) is additive and fully faithful.
The next theorem is the main result of [BS01].
Let C be an additive category, and let K be the bounded homotopy category of C.
Lemma A.8. Let A be an object in K with an idempotent endomorphism represented by a chain map p : A → A-in other words, p 2 is homotopic to p. For all odd n ≥ 1 there exist objects P n , Q n in K such that P n ⊕ Q n ≃ A ⊕ A[n], where p acts as the identity on P n and by zero on Q n .
Proof. We construct P n and Q n inductively. For n = 1 let P 1 := Cone(1 − p) and Q 1 := Cone(p). It is easy to see that they satisfy the desired properties. Assume that we constructed P n and Q n . We will construct P n+2 and Q n+2 as cones P n+2 = gn −→ Q n ] for certain chain maps f n and g n , which we will also construct.
Let us embed K into its Karoubi completion Kar(K). By Theorem A.7 the latter is triangulated. Since A has a homotopy idempotent p, we can split A ≃ P ′ ⊕ Q ′ for some objects P ′ , Q ′ in Kar(K) such that p acts by 1 on P ′ and by 0 on Q ′ . Now Similarly, Q 1 ≃ Q ′ ⊕ Q[1]. Observe that in Kar(K) there is a chain map f 1 : P 1 [1] → P 1 such that Cone(f 1 ) ≃ P ′ ⊕ P ′ [3]. Since the embedding K → Kar(K) is fully faithful, the map f 1 is well defined in K, and we can define P 3 := Cone(f 1 ). Similarly, if we already defined P n ≃ P ′ ⊕ P ′ [n] then in Kar(K) there is a chain map f n : P 1 [n] → P n such that P n+2 := Cone(f n ) ≃ P ′ ⊕ P ′ [n + 2].
Again, since the embedding is fully faithful the map f n (and hence P n+2 ) is well defined in K.
Analogously, one can define Q n such that Q n ≃ Q ′ ⊕ Q ′ [n] in Kar(K). Then P n ⊕ Q n ≃ A ⊕ A[n].
Remark A.9. One can write Since p(1 − p) vanishes up to homotopy, one can hope that the above sequences can be lifted to actual complexes by adding higher differentials. It is proved in [BN93, Propositions 3.1 and 3.2] that this construction is unobstructed.
Theorem A.10. The bounded homotopy category of a Karoubian category is Karoubian.
Proof. As above, let C be a Karoubian category and K its bounded homotopy category. Suppose that A is a complex in K with a homotopy idempotent p, we need to prove that A splits. Since A is bounded, we can pick a large enough odd positive integer n such that A and A[n] are supported in non-overlapping homological degrees. By Lemma A.8, one can decompose A ⊕ A[n] ≃ P ⊕ Q where p is homotopic to identity on P and to 0 on Q. Let us pick some homological degree i such that A is supported in degrees strictly smaller than i and A[n] is supported in degrees strictly bigger than i. Then (A ⊕ A[n]) i = 0.
Let h be a homotopy between p and identity on P . Since (A ⊕ A[n]) i = 0 we get dh i + h i+1 d = 1 as endomorphisms of P i . Let q = dh i and q ′ = h i+1 d, then q + q ′ = 1, q 2 = dh i dh i = (dh i + h i+1 d)dh i = dh i = q and similarly (q ′ ) 2 = q ′ . Therefore q and q ′ are orthogonal idempotents acting on P i , so (since C is Karoubian) we can rewrite P i = (P i ) ′ + (P i ) ′′ . Moreover, we can split P into two parts: P ′ = P <i → (P i ) ′ , and P ′′ = (P i ) ′′ → P >i . The same splitting works for Q. It is now easy to see that the map h <i induces a homotopy between (A⊕A[n]) ≤i = A and P ′ ⊕ Q ′ .
Remark A.11. Similarly to [BS01], one can instead deduce Theorem A.10 from Theorem A.7 and Theorem A.1. The proof presented here is slightly more explicit, following Proposition 1.5.6(iii) of [BV08].
Proposition A.12. Let C be a Karoubian monoidal category and let K be its homotopy category. Suppose that (E, s) is self-commuting in the sense of Remark 3.7. Then the Schur functors S λ (E) are well defined up to homotopy equivalence.
Proof. By the assumption, there is an action of S n on E ⊗n . If e λ is an idempotent in [S n ] then by Theorem A.10 there exists a splitting E ⊗n ≃ S λ (E) ⊕ E ′ where e λ acts by identity on S λ (E) and by 0 on E ′ . This splitting is unique up to isomorphism in K. Since {e λ } form an orthogonal system of idempotents, it is easy to see that E ⊗n ≃ |λ|=n S λ (E).