Almost-extreme Khovanov spectra

We introduce a functor from the cube to the Burnside 2-category and prove that it is equivalent to the Khovanov spectrum given by Lipshitz and Sarkar in the almost-extreme quantum grading. We provide a decomposition of this functor into simplicial complexes. This decomposition allows us to compute the homotopy type of the almost-extreme Khovanov spectra of diagrams without alternating pairs.

In 2018 the second author, together with Przytycki applied these techniques one step further, to the almost-extreme quantum grading. They restricted to the study of 1-adequate link diagrams, i.e., those whose 1-resolution contains no chords with both endpoints in the same circle (the extremal homology of these diagrams has been computed in [PS14,SS20,DL20]), and built a pointed semi-simplicial set whose homology is isomorphic to the almost-extreme Khovanov homology of the diagram: Theorem ( [PS20]). If D is a 1-adequate link diagram, then (i) The functor F j almax D gives rise to a pointed semi-simplicial set.
(ii) The realization of F j almax D has the homotopy type of a wedge of spheres and possibly a copy of a (de)suspension of RP 2 .
Their construction relates to the functor of Lawson, Lipshitz and Sarkar as follows: the category of pointed sets includes into the Burnside 2-category, and if a functor from the cube to the Burnside 2-category factors through it, then it gives rise to a pointed (augmented) semi-simpicial set.
Simplicial complexes and pointed semi-simplicial sets are simpler objects than functors to the Burnside 2-category, to which many classical tools can be applied, so these results prompt the question Are there simpler models of the spectra of Lipshitz and Sarkar in the almost-extreme quantum grading?
Unfortunately, it turns out that F j almax D gives rise to a pointed semi-simplicial set if and only if D is 1-adequate. However, in this paper we overcome this problem by introducing a new functor M D , whose realization coincides with that of F j almax D , and gives rise to an (augmented) pointed semi-simplicial set in a broader number of cases. This is achieved by constructing a natural transformation between M D and F j almax D and proving the following result: Theorem A. The natural transformation is a homology isomorphism, and therefore the realizations of M D and F j almax D are homotopy equivalent.
The functor M D can be understood as a "change of basis" of F j almax D : The relevant generators of the Khovanov chain complex in its almost-extreme degree (which is the same as the chain complex of F j almax D ) are indexed by the circles of the state resolutions of D, whereas the generators of the chain complex associated to M D are indexed by the edges and connected components of certain graph whose vertices are the circles of the state resolutions. A crucial property at the almostextreme quantum grading is that the relevant involved graphs are forests, and therefore both basis have the same size (as it should be).
The functor M D is simpler than F j D in the sense that it takes values in morphisms sending generators to (multiples of) generators, whereas F j D takes values in morphisms sending generators to sums of possibly different generators.
This simplified new basis allows us to decompose M D in terms of independence simplicial complexes of simpler link diagrams (Proposition 7.5 and Remark 7.6). The last part of the paper is devoted to make explicit computations using this decomposition.
One of the advantages of M D over F j almax D is that it gives rise to an augmented pointed semi-simplicial set if and only if the 1-resolution of the diagram contains no alternating pairs (i.e., two chords whose endpoints alternate along the same circle). Moreover, the forementioned decomposition allows us to compute the homotopy type of the realization of M D in these cases: Theorem B. Let D be a diagram whose 1-resolution contains no alternating pairs.
(i) The functor M D gives rise to an augmented pointed semi-simplicial set.
(ii) If D is not 1-adequate then the realization of M D is homotopy equivalent to a wedge of spheres.
In [PS18] the second author conjectured (and proved for several cases) that the extreme Khovanov spectrum is always a wedge of spheres. If this conjecture were true, then the above decomposition would give an upper bound for the cone length of the spectrum X j almax D (Remark 7.8). This bound seems to be far from optimal, as the computations of this paper and [PS20] produce spectra of cone length at most 2, so we rise the following question: Are there diagrams whose almost-extreme Khovanov spectrum is not homotopy equivalent to a wedge of spheres and possibly (de)suspensions of projective planes?
The structure of the paper is as follows. In Sections 2 and 3 we recall the categorical notions that will be used along the paper. In Section 4 we develop some results on link diagrams which allow, in Section 5, to introduce the functor M D and prove Theorem A. In Section 6 we prove the first part of Theorem B. In Section 7 we decompose M D in terms of independence simplicial complexes and present three skein sequences which allow us to determine, in Section 8, the homotopy type of M D for all links with no alternating pairs, proving the second part of Theorem B.
S S i i and the identity on a set X is the span X Id ← X Id → X.
Remark 2.2. Note that a span s = (Q, s, t) from X to Y is determined by the matrix whose entries are the sets s x,y := s −1 (x) ∩ t −1 (y), for every x ∈ X, y ∈ Y . Giving a fibrewise bijection τ from a span s to a span s is the same as giving, for each x ∈ X and each y ∈ Y , a bijection τ x,y from s x,y to s x,y . The composition of s : X → Y and s : Y → Z is the span s with s x,z = y∈Y s x,y × s y,z . The span s is determined by the formal sum s(x) = y∈Y s x,y · y, for each x ∈ X. If a coefficient s x,y is not specified, we understand that s x,y = {x} (for example, if s(x) = y + z, then s(x) = {x} · y + {x} · z).
The category of finite sets Set can be mapped into the category of pointed finite sets Set • by sending X to X + := X ∪ { * }, and a morphism f : X → Y to the morphism f + that coincides with f on X and sends the basepoint of X + to the basepoint of Y + . Moreover, the category Set • sits inside B by sending a pointed finite set X to X { * }, and a morphism f : X → Y to the span The inclusions Set → Set • → B induce equivalences between the following 2-subcategories of B: • The essential image of Set • in B is the 2-subcategory of free spans.
• The essential image of Set in B is the 2-subcategory of very free spans. Fix now a commutative ring R with unit, and let R-Mod be the category of modules over R. There is a functor (2.1) A : B → R-Mod sending the finite set X to the free R-module R X , and a span s : X → Y to the homomorphism f : R X → R Y given, for every x ∈ X, by Note that every pair of spans connected by a 2-morphism are sent to the same homomorphism, and therefore the functor A is well-defined.

Cubes of pointed sets and augmented semi-simplicial pointed sets
The n-dimensional cube 2 n is the partially ordered set (poset) whose elements are n-tuples u = (u 1 , . . . , u n ) ∈ {0, 1} n endowed with the standard partial order so that u v if u i v i for all i. Following [LLS20], we regard 2 n as the category whose objects are the elements of this poset and, for any two such elements u, v, the morphism set Hom(u, v) has a single element ϕ u,v if u v, and is empty otherwise. This category has an initial element 1 = (1, 1, . . . , 1) and a terminal element 0 = (0, 0, . . . , 0).
The poset map | · | : 2 n → N given by |u| = i u i assigns a grading to each vertex in the cube. We write u v if u > v and |u − v| = 1, and we write u i v if, additionally, u and v differ in the ith-coordinate, i.e., if u i = 1, v i = 0 and u j = v j for i = j.
Given n ∈ Z, n −1, the finite ordinal [n] is the linearly ordered set {0 < 1 < . . . < n}. The augmented semi-simplicial category, ∆ inj * , has as objects these finite ordinals, and as morphisms injective order-preserving maps between them. The inclusion ∂ i : [n − 1] → [n] that forgets the i-th element is called ith-face map.
Definition 3.1. Given a category C, let C op denote its opposite category, obtained by reversing the morphisms. An n-dimensional cube of pointed (finite) sets is a functor F : 2 n → Set • , while an augmented semi-simplicial pointed (finite) set is a functor X : ∆ op inj * → Set • . As usual, we write X n and ∂ i for X([n]) and X(∂ i ), respectively.
These two categories are related by a functor λ : 2 n → ∆ op inj * which maps every vertex of the cube u to the ordinal [|u| − 1], and every morphism u i v to the opposite of the j<i u i th-face map.
Taking left Kan extension along the functor λ : 2 n → ∆ op inj * defines a functor , whose value on a cube of pointed sets F is defined explicitely as follows: its set of k-simplices is Λ(F ) k = |u|=k+1 F (u), for every u ∈ 2 n , and the ith-face map is the vertex of the cube 2 n obtained by replacing the (i + 1)th one in u by a zero.
Recall from [LLS17,Definition 4.1] that a strictly unitary lax 2-functor F from the cube 2 n to the Burnside category B consists of the data: satisfying that, for every u > v > w > z, the following diagram commutes: In practice, when defining a strictly unitary lax 2-functor, we use the following result, from [LLS17, Section 4]. We keep the notation u • v for the morphism u v: Lemma 3.2. A strictly unitary lax 2-functor from the cube category 2 n to the Burnside category B is uniquely determined (up to natural isomorphism) by the following data: satisfying the following two conditions: • , the following commutes:

Realizations and totalizations.
Let C be a model category, as the category of (pointed) topological spaces Top (resp. Top • ), the category of spectra Sp, or the category Ch(R) of chain complexes of R-modules, with R a commutative ring.
The totalization of a functor F : 2 n → C is the object of C given by This defines a functor Tot : On the other hand, the relative realization of a functor X : ∆ op inj * → C is the object in C given by A semi-simplicial object X : ∆ op inj → C may be seen as an augmented semi-simplicial set by defining X −1 = * , the final object of C. Its relative realization is the suspension of the realization of X: X ∼ = Σ|X|.

3.2.
From Burnside cubes to chain complexes. Let R be a ring, and let Ch(R) be the category of chain complexes of R-modules. There is a functor where the first functor is A, defined in (2.1), and the second functor sends an R-module to the chain complex given by the R-module concentrated at degree 0. We define the functor C * (−; R) as the composition where the first functor is the result of composing a 2-functor F : 2 n → B from B 2 n with K. Its value at the Burnside cube F can also be described as the chain complex whose k-cochains are and whose differential is Observe that since B → R-Mod sends 2-morphisms to identities, a natural transformation between Burnside cubes induces a morphism of chain complexes.
3.3. From Burnside cubes to spectra. In [LLS20], Lawson, Lipshitz and Sarkar gave an explicit construction of the totalization functor for cubes in the Burnside category. They associated, to each cube F : 2 n → B, a spectrum Tot F , and to each map f : F → G of Burnside cubes, a map Tot f : Tot F → Tot G, both well-defined up to homotopy. Additionally, the homology H * (Tot F ; R) was isomorphic to the homology of C * (F ; R), and the map induced by Tot f on homology was the one induced by C * (f ; R).
Remark 3.4. If C * (f ; R) is a homology isomorphism then Tot f is a homotopy equivalence of spectra.
On the other hand, in [CMS20] the authors showed that if a cube F : 2 n → B factors through some functorF : 2 n → Set • , then Tot F = Σ ∞ TotF , where the homotopy colimit is taken in pointed topological spaces. Therefore, by (3.2), we have: Proposition 3.5. If a Burnside cube F : 2 n → B in the Burnside category factors through Set • , then the pointed semi-simplicial set Λ(F ) satisfies Remark 3.6. The spectrum Tot F is denoted |F | in [LLS20]. The chain complex C * (F ; R) is denoted Tot F in [LLS17]. We reserve the notation Tot for the totalization of a cube, · for the relative realization of an augmented semi-simplicial object and | · | for the realization of a semi-simplicial object.

Knots and graphs
4.1. States. Let D be an oriented link diagram with n ordered crossings {c 1 < c 2 < . . . < c n }, where n + (n − ) of them are positive (negative). A (Kauffman) state of the diagram D is an assignation of a label, 0 or 1, to each crossing in D. The order of the crossings induces a bijection between the set of states of D and the elements of 2 n by considering u ∈ 2 n as the state that assigns the label u i to c i .
Smoothing a crossing c i of D consists on replacing it by a pair of arcs and a segment connecting them; the way we smooth the crossings depends on its associated label u i , as shown in Figure 1. The result of smoothing each crossing of D according to its label is the chord diagram D(u), a collection of disjoint circles together with some segments that we call 0-and 1-chords, depending on the value of the associated u i . We represent 1-chords as light segments, and 0-chords as dark ones. • O i (u) denotes the set of circles containing an endpoint of e i (u) (thus |O i (u)| is either equal to 1 or 2). If |O i (u)| = 2 (resp. |O i (u)| = 1), we say that e i (u) is a bichord (resp. monochord ). Given a state u of D, we define its associated state graph, G(u), as the labelled graph obtained by collapsing each circle of D(u) to a vertex so that each chord in D(u) becomes an edge in G(u); each edge inherits a label, 0 or 1, from the associated chord. The circles and chords of D(u) are in bijection with the vertices and edges of G(u). Therefore, the above notation introduced to refer to the circles and chords of D(u) will be used to refer to the vertices and edges of G(u). In particular, the set of vertices of G(u) is Z(u) and the set of edges labeled by 0 and 1 are E(u) and E(u), respectively.
In this setting, write G(u) for the subgraph obtained after removing the 1-edges from G(u). See Figure 2 for such an example. We consider loops as length-1 cycles.
Recall that given u, v ∈ 2 n , we write u i v if both vectors are equal in all but the i th coordinate, where u i = 1 and v i = 0. The partial order > is defined as the transitive closure of the above relation.
If u i v, then D(v) is obtained from D(u) by doing surgery on D(u) along the 1-chord e i (u) and adding a new chord e i (v). Observe that, depending on the cardinality of O i (u), there are two possible surgeries: • If |O i (u)| = 2, then |O i (v)| = 1 and D(v) is obtained from D(u) by joining two circles into one. The graph G(v) is obtained from G(u) by identifying the two vertices in O i (u) and adding a loop (the edge e i (v)) based on the identified vertex, as shown in Figure 3(a). We say that u i v a merging.  • If |O i (u)| = 1, then |O i (v)| = 2 and D(v) is obtained from D(u) by splitting one circle into two. In this case, G(u) is obtained from G(v) by collapsing the edge e i (v), as shown in Figure 3(b). We say that u i v is a splitting.
Remark 4.1. Given two states u and v of D so that u > v, with |u| − |v| = k, there exist k! possible chains connecting them. However, the total composition of the one-by-one surgeries induced by each of the possible chains does not depend on the chosen chain.
Given a surgery u i v, we compare G(u) and G(v): Definition 4.2. Given a state u ∈ 2 n with |u| = n − k and a chain (4.1) 1 = u 0 i1 u 1 i2 . . . i k u k = u connecting u to the state 1, we define Φ(u) as the number of loops in {e ij (u j )} k j=1 . In other words, Φ(u) counts the number of mergings in the chain.
We prove now that when Φ(u) = 1 and |π 0 G(u)| = |π 0 G( 1)|, the chords of c 1 adjacent to the circle z alternate with those of c 2 along the boundary of z.
Consider a chain as (4.1), and let u r−1 ir u r be the states right before and right after the merging is performed, respectively. Write O ir (u r−1 ) = {z 1 , z 2 } and define ε as the (possibly empty) set containing those edges e ij in the chain so that O ij (u r−1 ) = {z 1 , z 2 }, for r < j k.
Since Φ(u r−1 ) = 0, we have just shown that G(u r−1 ) consists on |Z( 1)| disjoint trees. When passing from u r−1 to u r , vertices z 1 and z 2 are identified and the 1-edge e ir becomes a 0-loop (i.e., a length-one cycle) in G(u r ). Figure 4(a)-(b) illustrates this process.
The case (2a) corresponds to the case when z 1 and z 2 belong to different connected components of G(u r−1 ). Since no more mergings are possible, no more cycles are created. Moreover, the length of the (unique) cycle c 1 in G(u) is |ε| + 1 (see Figure 4(c)).
The case (2b) corresponds to the case when z 1 and z 2 belong to the same connected component: the fact that there is a path connecting z 1 and z 2 in G(u r−1 ) implies that an additional cycle c 2 is created when identifying both vertices. As illustrated in Figure 4

Enhanced states.
An enhacement of a state u ∈ 2 n is a map x assigning a label +1 or −1 to each of the circles in Z(u); we note by (u, x) the associated enhanced state. Write Z + (u, x) and Z − (u, x) for the subsets of elements of Z(u) labeled by +1 and −1, respectively. Define, for the enhanced state (u, x), the integers which are the homological and quantum gradings for Khovanov homology, respectively [LS14]. Let x + be the constant enhacement with value +1 and, for a given circle z ∈ Z(u), let x + z be the enhacement assigning a positive label to every circle but z. Sometimes we will write u + = (u, x + ). Define is an enhanced state of D} and j almax (D) = j max (D) − 2; we refer to these numbers as extreme and almostextreme (quantum) gradings for Khovanov homology of the diagram D. It turns out that Then, Proof. From the definition of quantum grading, it holds that Relation (4.2) completes the proof. Corollary 4.5. Let (u, x) be an enhanced state of D satisfying q(u, x) = j almax (D).
Since we are interested in studying the almost-extreme Khovanov complex of a link diagram, in the next section we study the characterization of those states u so that Φ(u) equals 0 or 1.

4.3.
States with Φ(u) = 0, 1. We are interested now in characterizing those states taking part in the almost-extreme Khovanov complex, i.e., given a state u ∈ 2 n of D, we want to determine whether there exists an enhacement x so that q(u, x) = j almax (D), just by looking at D( 1).
Definition 4.6. Given u ∈ 2 n a state of D and a, b, c, d chords in D(u), we say that: ( Given v > u states of D, we write D(v) u for the chord diagram having the same circles as D(v), but only the 1-chords (1) Two non-parallel bichords.
The proof of the necessary condition (i.e., the implication ⇐) is a case-by-case straightforward checking. We use Lemmas 4.9 and 4.10 in the proof of the sufficient condition.
Lemma 4.9. Let w v > u be three states with Φ(w) = Φ(v). Then: (1) If D(v) u contains a bichord, then D(w) u contains a bichord or an alternating pair. Proof. Since Φ(w) = Φ(v), w i v is a splitting for some monochord e i (w). The lemma follows from Remark 4.7 together with the following facts (i = j = k = l): -If e j (v) is a bichord, then either e j (w) is a bichord or (e j (w), e i (w)) form an alternating pair.
is an alternating triple, or (e i (w), e j (w), e k (w), e l (w)) is a mixed alternating pair. -If (e j (v), e k (v), e l (v), e m (v)) is a mixed alternating pair, then (e j (w), e k (w), e l (w), e m (w)) is a mixed alternating pair. -If (e j (v), e k (v)) is an alternating pair and e l (v) is a bichord, then either (e j (w), e k (w)) is an alternating pair and e l (w) is a bichord or (e i (w), e j (w), e k (w), e l (w)) is a mixed alternating pair, or (e i (w), e l (w)) and (e j (w), e k (w)) are disjoint alternating pairs.
(2) If D(v) u contains an alternating pair, then D(w) u contains an alternating triple or an alternating pair and a bichord.
The lemma follows from Remark 4.7 together with the following facts (i = j = k = l): -If e j (v) is a bichord, then e j (w) and e i (w) are non-parallel bichords.
-If (e j (v), e k (v)) is an alternating pair, then either (e j (w), e k (w)) is an alternating pair and e i (w) is a bichord, or (e i (w), e j (w), e k (w)) is an alternating triple.
Proof of the sufficient condition of Proposition 4.8. Fix a chain Consider first the case Φ(u) > 0 and write m for be the minimal number such that u m−1 im u m is a merging. Therefore, e im (u m−1 ) ∈ D(u m−1 ) u is a bichord and applying Lemma 4.9 recursively, we deduce that D( 1) u contains a bichord or an alternating pair.
Consider now the case Φ(u) > 1 and write m < p for the first two indices such that u m−1 im u m and u p−1 ip u p are mergings. Therefore, Φ(u m ) = Φ(u p−1 ), and applying recursively Lemma 4.9 to the bichord e ip (u p−1 ) ∈ D(u p−1 ) u we deduce that D(u m ) u contains a bichord or an alternating pair. Apply once Lemma 4.10 and deduce that D(u m−1 ) u contains at least one of the following: two non-parallel bichords or an alternating triple or an alternating pair and a bichord.
Finally, applying in each case recursively Lemma 4.9 we deduce that D( 1) u contains at least one of the following configurations: (1) Two non-parallel bichords.
The chords a 1 , b 1 , a 2 , b 2 divide the circle z into four arcs (see Figure 6). We define the ladybug set L(u) of D(u) as the set whose elements are the two arcs of z that are reached by traveling along any of the chords a 1 , b 1 , a 2 , b 2 until z, and then taking a right turn. Remark 4.11. In the case when |π 0 G(u)| = |π 0 G( 1)| and a i = b i for i = 1, 2, the set L(u) coincides with the right pair giving rise to the celebrated ladybug matching defined in [LS14].
, all elements in Λ are loops and none pair among them constitute an alternating pair.
By Lemma 4.3(2b), G(u) contains two cycles c 1 and c 2 sharing a common vertex z u . Write z v for the circle in D(v) with the same property. Then z v is obtained from z u by performing surgery along the loops in Λ with their endpoints in z u .
Write L(u) = {P, Q} and let Λ P (resp. Λ Q ) be the subset of loops of Λ so that at least one of their endpoints lies in P (resp. Q). The disposition of c 1 and c 2 (stated in Lemma 4.3(2b)) implies that Λ P ∩ Λ Q = ∅. Define the following subsets of The endpoints of each loop e i (u) ∈ Λ P (resp. e i (u) ∈ Λ Q ) separate z u into two arcs: write d i for the unique arc in z u which is disjoint from Q (resp. from P ). Define the following subsets of R 2 : As there are no alternating pairs in Λ, we deduce that As a consequence, we can consider two disjoint open subsets A, B of R 2 such that Each of the arcs of L(v) is obtained by doing surgery on each of the arcs of L(u).
The surgery performed on P is supported in A, while the surgery perfomed on Q is supported in B. Since A ∩ B = ∅, it is possible to define P (resp. Q ) as the arc of L(v) obtained from P (resp. Q). Thus, P intersects P , Q intersects Q and P ∩ Q = Q ∩ P = ∅, as desired.
4.5. The bijection ρ u,v . Given u i v so that Φ(u) = Φ(v) = 1, then |π 0 G(u)| = |π 0 G(v)|, and therefore it is possible to define a bijection ρ u,v : L(u) → L(v) in the following way:  Figure 7). The bijection ρ u,v : L(u) → L(v) is given by sending P u to P v and Q u to Q v . In particular, in those cases when none of the endpoints of the chord e i (u) lie in P u nor Q u , ρ u,v becomes the identity. The following lemma follows immediately from Lemma 4.12: Lemma 4.13. Let u v, v w so that Φ(u) = Φ(w) = 1. Then the following diagram commutes  without loss of generality that P ∩ z 1 = ∅ and Q ∩ z 2 = ∅ as subsets of R 2 . Then, we define the bijection µ i (u) : Figure 8).
The next result is a consequence of Lemma 4.12.
Lemma 4.14. Let u i v j w, u j v i w. Then, the following squares commute:

Khovanov functors
In this section we review the functor given by Lipshitz and Sarkar in [LLS17,LLS20] and introduce a new Khovanov functor, giving a natural transformation between them which allows us to prove that the geometric realizations of both functors are homotopy equivalent at the almost-extreme quantum grading (Corollary 5.7). First, we introduce some notation.
Let u be a state of a diagram D so that Φ(u) = 0. Orient G(u) by fixing, for each of its edges, one of its two possible orientations. We define, for each e ∈ E(u), the subgraph e + ⊂ G(u) as the connected component of G(u) {e} towards which e is pointing. Note that e + is well defined, since Lemma 4.3 guarantees that G(u) has no cycles.
Given any two states u > v of D, there is an inclusion of the associated graphs G(u) ⊂ G(v), and we define the maps observe that the first one is an isomorphism if Φ(u) = Φ(v), while the second one is always injective. Additionally, if u i v, there exists a Burnside morphism 5.1. The Khovanov functor in almost-maximal grading. The key piece in the construction of the Khovanov spectra in [LLS20] was the Khovanov functor, whose associated stable homotopy type is a link invariant. Using Lemma 3.2, we restate now this functor, adapted to the particular case of the almost-maximal quantum grading. Given a link diagram D with n ordered crossings, consider the functor 2 defined, in a vertex u ∈ 2 n , as 3 : On a morphism ϕ u,v , with u i v, the span F u>v := F (ϕ u,v ) : F (u) → F (v) is given, for z ∈ Z(u) and u + = (u, x + ), by Finally, we specify 2-morphisms: Let u i v j w and u j v i w. We need to produce a 2-morphism F u,v,v ,w between the 1-morphisms F v>w • F u>v and F v >w • F u>v .
Consider first the case when Φ(u) = 0. If Φ(u) = Φ(v) = Φ(v ) = 0, Φ(w) = 1, then the chords e i (u) and e j (u) form an alternating pair attached to some circle In the case z = z 1 , label the circles involved in the mergings and splittings as Therefore, Then, the 2-morphism F u,v,v ,w consists of a bijection between O j (v) and O i (v ) given by the ladybug matching If Φ(u) = 0 and we are not in the previous situation, then every summand in the formal sums (F v>w • F u>v )(z) and (F v >w • F u>v )(z) has a singleton as coefficient, so there is a unique choice for the 2-morphism F u,v,v ,w .
In the case when Φ(u) = 1, then to the formal sum w + which has a singleton as coefficient. The (strictly unitary lax) 2-functor F defined above is the Khovanov functor given in [LLS20] at the almost-maximal quantum grading and, if D has n − negative crossings, the almost-extreme (maximal) Khovanov spectrum is defined as The original functor F from [LLS20] splits into functors F j which associates to each state u a set E u,j of enhancements so that q(u, x) = j, for every x ∈ E u,j . When particularizing to the case when j = j almax , Corollary 4.5 implies that if Φ(u) = 0, then the set E u,j can be identified with Z(u), whereas if Φ(u) = 1 then E u,j = {u + } .

5.2.
A new equivalent functor. We introduce now a (strictly unitary lax) 2functor M : 2 n → B, with the property that its realization coincides with the realization of F , as will be show in Corollary 5.7. As before, we start by defining 4 M for the vertices of the cube, corresponding to the states of D:  Finally, given u i v j w and u j v i w, we define the 2-morphism M u,v,v ,w between the 1-morphisms M v>w • M u>v and M v >w • M u>v as follows.
First, note that M u,v,v ,w is trivially defined when Φ(u) = Φ(v) or |L(w)| = 1, since it is a bijection between singletons. Then, we just need to specify the cases when Φ(u) = 0, Φ(w) = 1 and |L(w)| = 2, depicted in Figure 10. Moreover, the value of M v>w • M u>v and M v >w • M u>v on a component C ∈ π 0 G(u) equals the empty set or a singleton, unless f u,w (C) is the component of G(w) containing two cycles; a similar reasoning applies for edges e ∈ G(u), unless e (or possibly g u,v (e) or g u,v (e)) points to the two circles involved in the unique merging. We define M u,v,v ,w in such components and edges:  (c) If Φ(u) = 0 and Φ(v) = Φ(v ) = Φ(w) = 1 and |L(w)| = 2, then e i (w) and e j (w) are parallel bichords (see Figure 10(c)) and we have: In order to prove condition (C2), we need to show that 2-morphisms in this cube commute. To do so, we study the bijections obtained when we move along the cube. More precisely, starting from M 001>000 • M 011>001 • M 111>011 we move to M 010>000 • M 011>010 • M 111>011 and continue moving along the cube following  To simplify notation, write ρ i for the morphism ρ u,v if u i v; then, the commutative diagram of Lemma 4.13 associated to u i v j w and u j v i w becomes: We just need to study the cases when Φ(111) = 0, Φ(000) = 1 and |L(000)| = 2 (otherwise, the coefficients involved in all compositions of 1-morphisms of the edges of the cube are singletons, and therefore the 2-morphisms commute trivially). In principle, up to permutation, there are 8 different situations attending to the value of Φ in each vertex, depicted in Figure 12. However, it is not hard to check that situations (a), (e) and (f ) leads to inconsistency in the chord diagrams, and therefore they are not possible. We study the remaining 5 possible situations: (1) Figure 12 The composition of the 2-morphisms is the following square, which is a particular case of (5.4): The composition of 2-morphisms is the boundary of the union of two squares of the form (5.4) along an edge, which commutes:  Once we have defined M , we will prove that its geometric realization is equivalent to that of the Khovanov functor F . To do so, we introduce a natural transformation γ : M → F and show that the induced homomorphism γ * : C * (M ; Z) → C * (F ; Z) is in fact an isomorphism (see Section 3.2).
Given a state u ∈ 2 n , we set the natural natural transformation γ as follows: Next, for each u i v, we define a 2-morphism γ u,v : F u>v • γ u → M u>v • γ v , i.e., γ u,v makes the following diagram commute: If Φ(u) = 0 and Φ(v) = 0, γ u,v is defined as the identity: In the case when Φ(u) = 0 and Φ(v) = 1 we define γ u,v as µ i (u): And when Φ(u) = 1 = Φ(v), γ u,v is defined as the identity: Lemma 5.5. The natural transformation γ u is well defined.
Proof. We need to prove that γ u satisfies conditions (C1) and (C2) in Lemma 3.2. Condition (C1) follows from definition of γ u . Next, we prove (C2) by showing that 2-morphisms in the following cube commutes, similar as we did in proof of Lemma 5.4: The commutativity is clear when all coefficients are singletons, i.e., it holds unless Φ(u) = 0, Φ(w) = 1 and |L(w)| = 2; thus, we have to study the three following situations 5 , corresponding again to the cases illustrated in Figure 10: Now, for each of the three cases above, we have to study the bijections obtained when we move along the cube: starting from γ w • M v>w • M u>v , we move to F v>w • γ v • M u>v , and continue moving along the cube following Figure 11, until we reach again γ w • M v>w • M u>v .
We just need to consider the action of those compositions on the component C ∈ π 0 G(u) such that f u,w (C) ∈ π 0 G(w) contains two cycles (otherwise, the value of F v>w • γ v • M u>v is trivial, as explained in the proof of Lemma 5.4).
Consider the case (5.5), where edges e i (u) and e j (u) form an alternating pair (see Figure 10(a)). Then F v>w • γ v • M u>v (C) equals L(w) · w + , and we obtain the following values when moving along the cube as in Figure 11: We focus now in case (5.6), where e j (u) is a loop and e i (u) a bichord ( Figure  10(b) shows an example of this situation). In this case F v>w • γ v • M u>v (C) equals L(v) · w + , and we get: Consider the third case (5.7), where e i (u) and e j (u) are parallel bichords, so Figure 10(c)). Then F v>w • γ v • M u>v (C) = L(v) · w + , and we obtain: Hence, showing the commutativity of the cube reduces to proving the following equalities in the corresponding situations: The first line holds on the nose, whereas the second and third are a consequence of the commutativity of the following squares, proved in Lemma 4.14: Proposition 5.6. γ * : C * (M ; Z) → C * (F ; Z) is an isomorphism of chain complexes.
Proof. Given a state u ∈ 2 n with Φ(u) = 0 and a vertex z ∈ Z(u) in G(u), write C z for the component of G(u) containing z and write E z (u) for the set of edges incident to z. For each edge e ∈ E z (u), set The inverse of γ * is: The above proposition together with Remark 3.4 yield to the following result: Corollary 5.7. The spectra Tot F and Tot M are homotopy equivalent.

Pointed semi-simplicial sets
Definition 6.1. A link diagram D is 1-adequate (resp. 0-adequate) if G( 1) (resp. G( 0)) contains no loops. D is said to be adequate if it is both 0-adequate and 1-adequate. If D is either 0-adequate or 1-adequate, it is called semiadequate. A link is said to be (semi)adequate if it admits a (semi)adequate diagram.  7.1. The simplicial complex I D . Given a link diagram D, in [GMS18] authors introduce a simplicial complex whose associated cohomology complex coincides with the Khovanov homology of the link in the minimal quantum grading. We restate that construction in terms of the maximal quantum grading j max .
Definition 7.1. Let D be a link diagram. Its associated Lando graph G L (D) is constructed from D( 1) by considering a vertex for every monochord, and an edge joining two vertices if the endpoints of the corresponding monochords alternate along the same circle. We define the independence complex 6 associated to D, I D , as the simplicial complex whose set of vertices is the same as the set of vertices of G L (D) and σ = (e 1 e 2 · · · e k ) is a simplex in I D if and only if the vertices e 1 , e 2 , . . . , e k are independent in G L (D), i.e., there are not edges in G L (D) between these vertices. Theorem. [GMS18] Let L be an oriented link represented by a diagram D with p positive crossings. Then The poset of faces of the independence complex I D is precisely the subposet of the cube 2 n of those states u for which Φ(u) = 0.
Given a set of vertices {v 1 , . . . , v n }, each simplicial complex on these vertices gives rise to a downwards closed subposet of 2 n (i.e., if a state u belongs to the subposet and u > v, then v belongs to the subposet too): its poset of faces. Conversely, every downwards closed subposet of 2 n is the poset of faces of some simplicial complex.
Let S ⊂ Set be the full subcategory on ∅ and a singleton, and let S p ⊂ Set • be the full subcategory on the basepoint and S 0 . Downwards closed subposets of the cube are in bijection with functors 2 n → S. Subposets of the cube are in bijection with functors 2 n → S p (see discussion in the final section of [CMS20], for example). The realization of a subposet of the cube is the desuspension of the totalization of its associated functor 2 n → S p ⊂ Set • . Thus, the realization of a simplicial complex X coincides with the realization of its poset of faces.
Given u ∈ 2 n , we writeū for the state of 2 n satisfyingū i = u i , for 1 i n.
Definition 7.2. The categorical dual of a downwards (upwards) closed subposet X ⊂ 2 n is the upwards (downwards) closed subposet X * ⊂ 2 n → S p given by u ∈ X * if and only ifū ∈ X.
The complement of a downwards (upwards) closed subposet X ⊂ 2 n is the upwards (downwards) closed subposetX ⊂ 2 n given by u ∈X if and only if u / ∈ X.
The complement of the categorical dual of a downwards closed set X is again a downwards closed subposet of the cube whose associated simplicial complex is the Alexander dual of X. In general |X| Σ|X| if X is downwards closed and |X * | is Spanier-Whitehead dual to |X|. Observe that (1) If |X| |A| ∨ |B|, then |X * | |A * | ∨ |B * |, (2) If |X| S k , then |X * | S n−k−2 , where n is the dimension of the cube.
Given a link diagram D, the upwards closed subposet of the cube given by those states u such that Φ(u) = 0, which we denote X D , corresponds to the functor F jmax : 2 n → Set • ⊂ B. Definition 7.4. Given a link diagram D with n crossings, we consider the following subposets of the cube 2 n : • X D : is the subposet of 2 n consisting of those states u so that Φ(u) = 0 (as defined in previous section).
, for any monochord e ∈ D( 1). • Y D : is the subposet of 2 n consisting of those states u so that Φ(u) = 1 and D( 1) u contains an alternating pair.
is the subposet of 2 n consisting of those states u so that Φ(u) = 1 and D( 1) u contains a bichord parallel to a given bichord b of D( 1).
Observe that each state (admitting an enhacement) in the almost-extreme complex of D belongs to one and only one of the previous subcubes X D , Y D or Z D (see Corollary 4.5 and Proposition 4.8).
Proposition 7.5. The functor M D can be realized as the following cofibre sequence where N is the set of monochords in G( 1) and B the set of classes of parallel bichords.
The decomposition follows because the second map is levelwise injective and the suspension of the leftmost term is precisely the quotient functor of this second map.
Thus, the (first page of the) Mayer-Vietoris spectral sequences associated to these coverings are: Example 7.7. Consider the torus knot T (3, q) and let D = D (3,q) be its standard diagram. Since D( 1) contains no bichords, Z b D is trivial. We will combine Remark 7.6 together with Corollary 7.3 to compute the realization of subposets in (7.1). More precisely, we will express X D , X e D and Y D (via A k ) as the duals of independence complexes of some Lando graphs. Write C n for the cycle graph of n vertices and L n the path of length n. Note that there are no n-tuples of monochords e 1 , . . . , e n which can be divided into two subsets H 1 and H 2 satisfying the condition above when n > 3. Summarizing we have: The homotopy types of the above complexes were computed in [PS18, Corollary 3.4 and Proposition 3.9]: This allows to compute the almost-extreme Khovanov spectra of T (3, q) in terms of well-known independence complexes of cycles and paths, with no need to apply induction on q.
Remark 7.8. The cone-length c(X ) of a spectrum X is the least n such that there is a sequence of cofibre sequences for i = 0, . . . , n − 1, such that X 0 is contractible, Y i is a wedge of spheres and X n = X . In [PS18] it was conjectured that |X D | is homotopy equivalent to a wedge of spheres for any diagram D. If this were true, then Remark 7.6 would imply that the cone length of Tot M D is bounded above by the maximum of the following numbers: (1) the maximum number of parallel bichords plus one in D( 1).
whereas if a is a bichord in D( 1), then the skein short exact sequence becomes Moreover, when a is a monochord in D( 1) we also have the following sequence: To verify the three above sequences note that the second map is an inclusion whose quotient is the suspension of the first spectrum.
Definition 7.9. Let D be a diagram and let a be a monochord in D( 1). We say that a is: (1) 2-free if it is not part of any alternating pair in D( 1).
(2) 3-free if it is not part of any alternating triple in D( 1).
(3) free if it is 2-free and 3-free.
(4) b-free if it is not part of any alternating triple involving the bichord b in D( 1).
Lemma 7.10. Let D be a diagram and let a be a monochord in D( 1).
(1) If a is 2-free, then X D , Y D and all X e D , with e = a a monochord, are contractible; if, additionally, D( 1) contains another 2-free monochord, then X a D is contractible too. (2) If a does not form an alternating triple with a bichord b (i.e., a is b-free), then Z b D is contractible. Proof. The leftmost map in skein sequence (7.4) along the crossing associated to a is identity, hence X D is contractible. A similar reasoning works for X e D when e = a. In general, the skein sequence (7.2) does not respect the decomposition in (7.1); however, when a is 2-free, it respects the term Y D and when a is b-free it respects Corollary 7.11. If D( 1) contains at least two 2-free monochords, then Proof. The result is direct from cofibre sequence (7.1) and Lemma 7.10(1). Proof. We use skein sequence (7.2) and show that M D[a=0] is contractible. First, notice that the circle in the statement is splitted into two circles z 1 and z 2 when changing the 1-label of a for a 0-label, each of them attached to at least a 2-free monochord that we call c and d, respectively. Lemma 7.10(1) implies that the the only possibly non-contractible spaces when applying cofibre sequence (7.1) to

Diagrams with no alternating pairs
In this section we determine the homotopy type of M D (thus, that of the Khovanov spectrum given by Lipshitz and Sarkar in [LS14] for the almost-extreme quantum grading) for diagrams D so that D( 1) contains no alternating pairs. Observe that this determines their Khovanov homology groups at almost-extreme quantum degree.  Proposition 8.1. Let D be a link diagram with n crossings so that D( 1) contains a single monochord attached to a circle z, and let k be the number of circles connected to z along an alternating triple. Then, Proof. Write e for the monochord in D( 1). By Lemma 7.10(1), the decomposition in (7.1) becomes The subposet X e D has a single element in degree n − 1, thus |X e D | S n−2 . By Lemma 7.10(2), Z b D is contractible unless b forms an alternating triple with e. Therefore we may assume, up to suspension, that all bichords are attached to z and form an alternating triple with e. Moreover, using Lemma 7.15 we can assume, up to suspension, that circles are connected to z by exactly two bichords.
Therefore, for each such bichord b, the subposet Z b D has five elements in M-shape, with two elements in degree n − 1 and three in degree n − 2, thus |Z b D | S n−3 , and the above decomposition becomes The map to the wedge of spheres is a diagonal map. This concludes the proof. 8.3. Diagrams with 2-free monochords. Let D be a diagram whose associated chord diagram D( 1) contains more than one monochord, all of them 2-free. Recall from Corollary 7.11 that in this situation M D [b]∈B Z b D , thus we can restrict to the independent study of each connected pair of circles when computing Z b D . At this point we can make some simplifications in D( 1): Lemmas 7.13 and 7.15 allow us to remove nested monochords and equivalent bichords when computing M D (in exchange of taking suspensions). Moreover, by Lemma 7.10(2) we can assume that D( 1) contains no b-free monochords for any bichord b. These simplifications motivate the following definition: contains at least two monochords, all of them 2-free, and satisfies the following conditions: (1) It contains exactly two circles; (2) It contains no nested monochords; (3) There are no b-free monochords for any bichord b; (4) There are no equivalent bichords.
By definition, if D is simple then D( 1) can be isotoped in S 2 so that monochords and bichords lie in different regions (see Figure 13). Moreover, since D( 1) contains no nested monochords, a circle with n monochords is divided into n half-disks (each of them bounded by a monochord and an arc of the circle) and an additional region (bounded by the n monochords and n arcs of the circle) that we call polygon.  ( 1) has a single circle and its inner part is separated into two regions R and R by the monochord e. See Figure 14 (2) |d e | = 2, and one bichord connects d e with a half-disk in the other circle and the other one connects d e with the polygon in the other circle (see Figure 15(b)); (3) |d e | = 3, and two bichords connect d e with two contiguous half-disks in the other circle, and the third one is placed between them and connects d e with the polygon of the other circle (see Figure 15(c)). The next statement follows from Lemmas 8.7 and 8.8.
Proposition 8.6. Let D be a super-simple diagram. Then |M D | is homotopy equivalent to a wedge of spheres.
Lemma 8.7. Let D be a super-simple diagram so that D( 1) contains at least one bichord connecting a half-disk with the region called polygon. Then |M D | is either contractible or homotopy equivalent to a sphere.
Proof. Let b be a bichord connecting a half-disk d e in a circle z 1 with the polygon of the circle z 2 , for some monochord e. See Figure 16(a). Since D is super-simple, there exists at least a bichord a connecting d e with d f for a monochord f attached to z 2 . We distinguish two cases, depending on whether z 2 contains at least two monochords or it contains just a single monochord.
Case 1: Assume that the chord diagram D( 1) contains more than one monochord attached to z 2 , and consider the skein exact sequence (7.3) along a: We will show that X D[a=0] is contractible. To do so, we apply the skein sequence (7.4) along the monochord f :  Case 2: Assume that there is just one monochord f attached to the circle z 2 in D( 1). Then, since D is super-simple, D( 1) is as one of the six chord diagrams depicted in Figure 17 (each picture leads to two possible chord diagrams, depending on whether it contains the bichord p or not). In order to compute M D for each of these situations, we use skein sequence (7.3) together with Corollary 7.3. We also use [PS18, Lemma 3.2, Corollary 3.3] to compute I D : -If D( 1) is as in Figure 17(a) with p: we consider the skein sequence (7. 3) along the bichord c, and get that |X D[c=0] | S 2 . Next, we show that M D[c=1] is contractible by applying the skein sequence (7.3) to D[c = 1] along the bichord a (we use the fact that chord f is free in D [c=1,a=1] ( 1)). Therefore, |M D | |X D[c=0] | S 2 .
-If D( 1) is as in Figure 17 Figure 17(b) without p: the procedure is analogous to the previous one, the only difference is that |X D[a=0] | S 3 and therefore |M D | S 3 . -If D( 1) is as in Figure 17(c) with p: we consider the skein sequence (7. 3) along the bichord a, and get that X D[a=0] is contractible. Next, we show that M D[a=1] is also contractible by applying the skein sequence (7.3) to D[a = 1] along the bichord b (we use the fact that chord e is free in D [a=1,b=1] ( 1)). Therefore, M D is contractible. Proof. First, notice that since all bichords in D( 1) connect two half-disks, the number of monochords equals the number of bichords. We label the chords as follows (see Figure 18(a)): we write e 1 , . . . , e n (resp. f 1 , . . . , f n ) for the monochords attached to the circle z 1 (resp. z 2 ), and a i (resp. b i ) for the bichord having its endpoints in the half-disks d ei and d fi (resp. d ei+1 and d fi ), for 1 i n.
Consider the skein sequence (7.3) along the bichord a 1 : We study now the homotopy type of X D[a1=0] . Figure 18(b) shows the chord diagram D[a 1 = 0]( 1), whose associated Lando graph G = G L (D[a 1 = 0]) is as depicted in Figure 18(c). Now, since vertex e 1 dominates 9 e 2 and vertex f 1 dominates f 2 , it follows from [PS18, Lemma 3.2] that the independence complex associated to G L is homotopy equivalent to the independence complex associated to the graph G − e 1 − f 1 , which is a path of length 4n − 4, L 4n−4 , whose independence complex follows from [PS18, Corollary 3.4]:    Theorem. Let D be a diagram so that the associated chord diagram D( 1) contains more than one monochord, all of them 2-free . Then |M D | is homotopy equivalent to a wedge of spheres.
Proof. Since all monochords are 2-free, then M D [b]∈B Z b D , by Corollary 7.11, so we consider each pair of discs in D( 1) independently. Moreover, we can remove nested monochords and equivalent bichords when computing M D at the expense of taking suspensions (Lemmas 7.13 and 7.15). In addition, we can assume that D( 1) contains no b-free monochords for any bichord b (otherwise, Z b D is contractible by Lemma 7.10(2)). Hence, we just need to prove the statement for simple diagrams. Lemma 8.3 and Proposition 8.6 complete the proof.