Quantum link homology via trace functor I

Abstract

Motivated by topology, we develop a general theory of traces and shadows for an endobicategory, which is a pair: bicategory and endobifunctor . For a graded linear bicategory and a fixed invertible parameter q, we quantize this theory by using the endofunctor $\Sigma _q$ such that $\Sigma _q \alpha :=q^{-\deg \alpha }\Sigma \alpha$ for any 2-morphism $\alpha$ and coincides with $\Sigma$ otherwise. Applying the quantized trace to the bicategory of Chen–Khovanov bimodules we get a new triply graded link homology theory called quantum annular link homology. If $q=1$ we reproduce Asaeda–Przytycki–Sikora homology for links in a thickened annulus. We prove that our homology carries an action of , which intertwines the action of cobordisms. In particular, the quantum annular homology of an n-cable admits an action of the braid group, which commutes with the quantum group action and factors through the Jones skein relation. This produces a nontrivial invariant for surfaces knotted in four dimensions. Moreover, a direct computation for torus links shows that the rank of quantum annular homology groups depend on the quantum parameter q.

Appendix A Background survey

The material presented here is widely known, and the main goal of this section is to fix the notation. Bicategories are treated in the excellent paper [11], whereas [34] is a brief list of basic definitions. The reader is also referred to [23], because many results about monoidal categories immediately translates to bicategories.

1.1 A.1 Representations of

As usual we fix a commutative unital ring together with an invertible element q. By definition, is the unital associative -algebra with generators E, F, K, $K^{-1}$ and relations

$$\begin{aligned} KE&= q^2EK,&KK^{-1}&= 1 = K^{-1}K, \\ KF&= q^{-2}FK,&K-K^{-1}&= (q-q^{-1})(EF-FE). \end{aligned}$$

It is a Hopf algebra with the comultiplication , the counit , and the antipode defined by

$$\begin{aligned} \Delta (E)&= E\otimes K+1\otimes E,&\epsilon (E)&= 0,&S(E)&= -EK^{-1},\\ \Delta (F)&= F\otimes 1+K^{-1}\otimes F,&\epsilon (F)&= 0,&S(F)&= -KF,\\ \Delta (K^{\pm 1})&= K^{\pm 1}\otimes K^{\pm 1},&\epsilon (K^{\pm 1})&= 1,&S(K^{\pm 1})&= K^{\mp 1}. \end{aligned}$$

Using this Hopf algebra structure, we can regard the category of finite-dimensional representations of as a monoidal category with duals. The unit in this monoidal category is given by the trivial representation , on which acts by multiplication by $\epsilon (X)$ for any .

We write and for the fundamental representation and its dual. We identify both with the rank two module using the isomorphisms

(A.1)

This equips V with two actions of that differ by signs:

The duality between $V_1$ and $V_1^*$ comes with the evaluation and coevaluation maps

that intertwine the action if $V\otimes V$ is identified with either $V_1\otimes V_1^*$ or $V_1^*\otimes V_1$.

The full subcategory of generated by tensor powers of V admits a graphical representation. Let be the Temperley–Lieb category, the linear category with objects finite collections of points on a real line and morphisms generated by flat loopless tangles, i.e. collection of disjoint intervals in $\mathbb {R}\times I$ with endpoints on the boundary lines. Composition is defined by stacking pictures one onto another and trading each closed component for $q+q^{-1}$. For example,

There is a functor that assigns $V^{\otimes n}$ to a collection of n points, whereas caps and cups are sent to the evaluation and evaluation homomorphisms. It is known that is faithful [47].

1.2 A.2 Knots and tangles

Let M be an oriented smooth 3-manifold. A proper 1-submanifold $T\subset M$ is called a tangle. We call it a link if it has no boundary, and a knot if in addition it has one component. All tangles and links in this paper are assumed to be oriented unless stated otherwise. An isotopy of tanglesT and $T'$ is a smooth map $\Phi {:}\,M\times I\rightarrow M$ such that each $\Phi _t := \Phi (-, t)$ is a diffeomorphism fixed at the boundary, $\Phi _0 = {{\,\mathrm{id}\,}}$, and $\Phi _1(T) = T'$. If T and $T'$ are oriented, then we require that the orientation is preserved by $\Phi _1$.

Denote by $-T$ the tangle T with reversed orientation of all its components. A tangle cobordism from a tangle $T_0$ to $T_1$ is an oriented surface $S\subset M\times I$ with boundary $\partial S = -T_0\times \{0\} \cup T_1\times \{1\} \cup (-\partial T_0\times I)$. We shall consider cobordisms only up to an isotopy, in which case they form a category: composition is given by gluing cobordisms, and the identity morphism on a tangle T is represented by the cylinder $T\times I\subset M\times I$.

When $M= F\times I$ is a thickened surface, then isotopy classes of oriented tangles in M form a category, with the product induced by stacking, $M\cup M \cong M$, and tangles with tangle cobordisms form a 2-category.

Notation

We shall write $Links (M)$ for the set of isotopy classes of oriented links in M, and for the category of oriented links in M and cobordisms between them. Isotopy classes of oriented tangles in a thickened surface $F\times I$ form a category with the composition induced by stacking, and similarly cobordisms between tangles in $F\times I$ form a 2-category $\mathbf {Tan}(F)$. We write simply and $\mathbf {Tan}$ when $F=\mathbb {R}\times I$.

Assume M is a line bundle over a surface $F$ and consider the projection $M\rightarrow F$ onto the zero section. It maps a generic tangle T to an immersed collection of intervals and circles $\tilde{T}\subset F$ with only finitely many multiple points, each a transverse intersection of two arcs. A diagram of T is constructed from $\tilde{T}$ by breaking one of the arcs at each double point as follows. When $F$ is oriented, then fibers of M admit a canonical orientation and we break the lower arc at each double point of $\tilde{T}$ (see Fig. 12 for the case $F=\mathbb {R}\times I$). In case $F$ is nonorientable, choose a minimal collection of curves $\gamma$ that cuts $F$ into an orientable surface. Then there is a normal field over $F- \gamma$ and we can construct the diagram as before. Then Reidemeister moves and planar isotopies relate diagrams between isotopic tangles if a crossing is switched when moved through $\gamma$:

Fig. 12

An example of a tangle diagram

This follows, because the normal field is reversed at points of $\gamma$.

1.3 A.3 Constructions on categories

Below we review definitions of certain constructions on categories that appear throughout the paper.

1.3.1 Additive closure

We say that a linear category is additive if it has finite direct sums. Each category admits the additive closure, the smallest additive category containing . It is constructed by introducing formal direct sums:

• an object of is a finite sequence $(x_1,\ldots ,x_r)$ with , possibly empty,

• a morphism from $(x_1,\ldots ,x_r)$ to $(y_1,\ldots ,y_s)$ is a matrix $(f_{ij})$ of morphisms , and

• composition is defined by the matrix multiplication rule.

There is an inclusion , which takes an object x to the 1-element sequence (x). It is an equivalence of categories if is already additive.

1.3.2 Idempotent completion

An endomorphism satisfying $p\circ p = p$ is an idempotent. We say that psplits if it decomposes $p = s\circ r$ such that $r\circ s$ is an identity morphism. In such a case r is an epimorphism and its codomain is called the image ofp.

A category is idempotent complete if all its idempotents split. Each category admits its idempotent completion, also called also the Karoubi envelope of, which is the smallest idempotent complete category containing . It is constructed by taking all idempotents of as objects, and defining morphisms from e to $e'$ as those morphisms f from that $e'\circ f\circ e$ is well-defined and equal to f. The identity morphism on e is given by the idempotent itself.

1.3.3 Formal complexes and homotopy category

A formal complexes over a category is a sequence of objects and morphisms from

$$\begin{aligned} (C^\bullet , d) = \left( \cdots \rightarrow C^i \rightarrow ^{d^i} C^{i+1} \rightarrow ^{d^{i+1}} C^{i+2} \rightarrow \cdots \right) \end{aligned}$$

(A.3)

satisfying $d^{i+1}d^i = 0$ at each place. The morphisms $d^i$ are called the differential. We say that $(C^\bullet , d)$ is bounded if $C^i = 0$ except finitely many indices.

A formal chain map from $(C^\bullet , d)$ to $(D^\bullet , d)$ is a collection of morphisms $f^i{:}\,C^i \rightarrow D^i$ fitting into a commuting ladder

(A.4)

Finally, a formal chain homotopy from $f^\bullet$ to $g^\bullet$, both chain maps from $(C^\bullet ,d_C)$ to ($D^\bullet , d_D)$, is a collection of morphisms $h^i{:}\,C^i\rightarrow D^{i-1}$ satisfying $d^{i-1}_D\circ h^i + h^{i+1}\circ d^i_C = g^i - f^i$. In such case we say that $f^\bullet$ and $g^\bullet$ are homotopic, which we write $f\sim g$.

Formal complexes (reps. bounded formal complexes) and chain maps modulo chain homotopies constitute the homotopy category of complexes (resp. ). Isomorphism in these categories are called homotopy equivalences and we usually write $C^\bullet \simeq D^\bullet$ for complexes that are homotopically equivalent.

The categories and are triangulated [24, 49], which means that they come with a homological degree shift functor and a collection of distinguish triangles satisfying certain axioms. The degree shift functor is usually denoted by [1] and it shifts a complex leftwards, negating the differential at the same time:

$$\begin{aligned} C[1]^i := C^{i+1}, \qquad d[1]^i := -d^{i+1}. \end{aligned}$$

(A.5)

Distinguished triangles are of the form

$$\begin{aligned} C^\bullet \rightarrow ^{f^\bullet } D^\bullet \rightarrow ^{ in ^\bullet } \mathrm {cone}^\bullet (f) \rightarrow ^{ pr ^\bullet } C[1]^\bullet , \end{aligned}$$

(A.6)

where $\mathrm {cone}^\bullet (f)$ stands for the mapping cone of $f_{}$, the formal complex

$$\begin{aligned} \mathrm {cone}^i(f^\bullet ) := C^{i+1}\oplus D^i, \qquad d^i = \begin{pmatrix} -d^{i+1}_C &{}\quad 0 \\ f^i &{}\quad d^i_D \end{pmatrix}. \end{aligned}$$

(A.7)

The morphisms $in ^i{:}\,D^i\rightarrow \mathrm {cone}^i(f^\bullet )$ and $pr ^i{:}\,\mathrm {cone}^i(f^\bullet )\rightarrow C[1]^i = C^{i+1}$ are the inclusion and projection respectively.

1.4 A.4 Bicategories

A bicategory$\mathbf {C}$ is a ‘higher level’ analogue of a category. It consists of

• a class of objects $\mathrm {Ob}(\mathbf {C})$,

• a category $\mathbf {C}(x,y)$ for each pair of objects (x, y), whose objects and morphisms are called 1- and 2-morphisms respectively and represented by single and double arrows,

• a unit ${{\,\mathrm{id}\,}}_x\in \mathbf {C}(x,x)$ for each object x,

• a functor $\circ {:}\,\mathbf {C}(y,z) \times \mathbf {C}(x,y) \rightarrow \mathbf {C}(x,z)$ for each triple of objects (x, y, z), and

• natural isomorphisms

  

  called associators and unitors, which satisfy the pentagon and triangle axioms [11].

A bicategory is called strict or a 2-category if the natural isomorphisms are identities. We call $\mathbf {C}$ a locally small bicategory when each category $\mathbf {C}(x,y)$ is small, and $\mathbf {C}$ is small when also $\mathrm {Ob}(\mathbf {C})$ is a set.

Associators and unitors are often omitted for clarity. According to the MacLane’s Coherence Theorem [36, Chapter VII.2] there is only one way how to insert these isomorphisms back when necessary.

Notation

In this paper we denote categories with calligraphic letters , , etc., whereas bold letters $\mathbf {C}$, $\mathbf {D}$, etc. are reserved for bicategories. Identity morphisms are written as ${{\,\mathrm{id}\,}}_x$, and identity 2-morphisms as $\mathbf 1_f$. If $\mathbf {C}$ is a bicategory, then the composition in $\mathbf {C}(x,y)$ is denoted by $*$ and called vertical, whereas $\circ$ is the horizontal composition. These come from the common convention to draw 1-morphisms horizontally and 2-morphisms vertically. For instance, a 2-morphism is commonly presented as

Choose bicategories $\mathbf {C}$ and $\mathbf {D}$. A bifunctor$\mathbf {F}{:}\,\mathbf {C}\rightarrow \mathbf {D}$ consists of a function of objects $\mathrm {Ob}(\mathbf {C}) \rightarrow \mathrm {Ob}(\mathbf {D})$ and a collection of functors $\mathbf {C}(x,y) \rightarrow \mathbf {D}(\mathbf {F}x,\mathbf {F}y)$, together with natural 1-morphisms

satisfying certain coherence axioms. They are called morphisms of bicategories in [11], whereas the word homomorphism is reserved for the case when $\mathfrak m$ and $\mathfrak i$ are invertible. In such case we say that $\mathbf {F}$ is a strong bifunctor.

Choose bifunctors $\mathbf {F},\mathbf {G}{:}\,\mathbf {C}\rightarrow \mathbf {D}$. A natural transformation$\eta {:}\,\mathbf {F} \rightarrow \mathbf {G}$ is a collection of 1-morphisms $\eta _x{:}\,\mathbf {F}(x) \rightarrow \mathbf {G}(x)$, one per object $x\in \mathbf {C}$, and 2-morphisms , one per 1-morphisms $f\in \mathbf {C}(x,y)$, such that

for every 2-morphism . Moreover, $\eta _f$ must be coherent with all the other canonical 2-isomorphisms (associators, unitors, the structure 2-isomorphisms of $\mathbf {F}$ and $\mathbf {G}$), see [34]. We say that $\eta$ is strong if each $\eta _f$ is invertible.

Finally, let $\eta ,\nu {:}\,\mathbf {F}\rightarrow \mathbf {G}$ be two natural transformations. A modification$\Gamma {:}\,\eta \rightarrow \nu$ is a collection of 1-morphisms $\Gamma _x{:}\,\eta _x\rightarrow \nu _x$, such that $\nu _f*({{\,\mathrm{\mathbf {1}}\,}}_{\mathbf {G}f}\circ \Gamma _x) = (\Gamma _y\circ {{\,\mathrm{\mathbf {1}}\,}}_{\mathbf {F}f})*\eta _f$ for every 1-morphism $f{:}\,x\rightarrow y$.

1.4.1 Duals

A bicategory $\mathbf {C}$ has left duals if each $f\in \mathbf {C}(x,y)$ admits ${}^*\!f\in \mathbf {C}(y,x)$ together with coevaluation and evaluation 2-morphisms

fitting into commuting triangles

where for clarity associators and unitors are omitted. The morphism ${}^*\!f$ is called the left dual to f. We define the right dual$f^*$ of f by reversing the order of the horizontal composition in (A.16) and (A.17). If a dual 1-morphism exists, then it is unique up to an isomorphism. In particular, dual pairs are preserved by strong bifunctors.

1.4.2 Examples

Small categories, functors, and natural transformations form a strong bicategory $\mathbf {Cat}$. A left (resp. right) dual to a functor F is its left (resp. right) adjoint. Hence, not all 1-morphisms in $\mathbf {Cat}$ are dualizable.

Tangles in a thickened surface $F\times I$ constitute a bicategory $\mathbf {Tan}(F)$, which objects are finite collections of points in $F$, 1-morphisms are tangles with endpoints on $F\times \partial I$, and 2-morphisms are tangle cobordisms. This bicategory has both left and right duals. Indeed, the mirror image $T^!$ of a tangle T, obtained by flipping $F\times I$, is both the left and right dual of T. The evaluation 2-morphism is obtained by revolving T in four dimensions along the input surface $F\times \{0\}$, i.e. it is the image of the map $(p,t,s)\mapsto (p,t\cos (s\pi ),t\sin (s\pi ))$ with $(p,t)\in T$ and $s\in I$, suitably normalized (see Fig. 13). The coevaluation is defined dually by a rotation along the output surface $F\times \{1\}$.

Fig. 13

The evaluation cobordism for a tangle $T\in \mathbf {Tan}(1,3)$

Rings, bimodules, and bimodule maps form a bicategory $\mathbf {Bimod}$, with horizontal composition given by the tensor product: for an (A, B)-bimodule M and a (B, C)-bimodule N. This formula comes from interpreting an (A, B)-bimodule M as a functor between the categories of right modules. This bicategory does not have duals. Indeed, an (A, B)-bimodule M has a left (resp. right) dual if and only if it is finitely generated and projective as a right B-module (resp. left A-module). If so, the left and right dual modules are given as the modules of right B-linear and left A-linear morphisms respectively:

For this reason we usually restrict either to $\mathbf {Rep}$ (bimodules with left duals) or $\mathbf {Birep}$ (bimodules with both left and right duals).