Bundles of coloured posets and a Leray-Serre spectral sequence for Khovanov homology

The decorated hypercube found in the construction of Khovanov homology for links is an example of a Boolean lattice equipped with a presheaf of modules. One can place this in a wider setting as an example of a coloured poset, that is to say a poset with a unique maximal element equipped with a presheaf of modules. In this paper we initiate the study of a bundle theory for coloured posets, producing for a certain class of base posets a Leray-Serre type spectral sequence. We then show how this theory finds application in Khovanov homology by producing a new spectral sequence converging to the Khovanov homology of a given link.


Introduction
Our motivation for studying coloured posets comes from a desire to understand better some of the structures arising in the Khovanov homology of links. One can ask if Khovanov's "cube" construction can be placed in a broader context, hoping that this new perspective allows one to lever some advantage. In [2] we showed that this is possible using the framework of what we call coloured posets. This is something of a half-way house between the specifics of "cubes" and the full generality of sheaves of modules over small categories. We believe this is an appropriate place to study Khovanov homology as it provides enough generality to be useful, without losing sight of the starting point. The idea being that one can then bring the algebraic topology of coloured posets into the game.
One direction of interest is a bundle theory for coloured posets which we initiate in this paper. Given a poset B with unique maximal element we define bundles of coloured posets over B. Roughly, we wish to capture the idea that a coloured poset may be decomposed as a family of coloured posets parametrized by another poset, much as a fibre bundle is a family of spaces parametrized by some base space. A bundle of coloured posets has a total coloured poset (E, F), and our central interest is in computing its homology. Our main result in this direction, proved as Theorem 1 in §5, is a Leray-Serre style spectral sequence for bundles where the base has a certain technical property that we call special admissibility: Main Theorem. Let ξ : B → CP R be a bundle of coloured posets with B finite and specially admissible, and let (E, F) be the associated total coloured poset. Then there is a spectral sequence E 2 p,q = H p (B, H fib q (ξ)) =⇒ H * (E, F). It turns out that a number of naturally occurring posets are specially admissible, and importantly for applications to knot homology theories, this class includes the Boolean lattices. Given a knot diagram D together with k distinguished crossings c 1 , . . . , c k , one can obtain a new complex in the spirit of Khovanov: resolving each of the remaining ℓ crossings gives another link diagram, and the ℓ diagrams that result can be placed as the vertices of a hyper-cube (a Boolean lattice). Taking the unnormalised Khovanov homology of these diagrams and using induced maps along edges one obtains, by the construction of Khovanov, a triply graded complex whose homology we denote KH * , * , * (D; c 1 , . . . , c k ).
Main Application. Let D be a link diagram and let c 1 , . . . , c k be a subset of the crossings of D. For each i, there is a spectral sequence E 2 p,q = KH p,q,i (D; c 1 , . . . , c k ) =⇒ KH p+q,i (D).
The r-th differential in the spectral sequence has bidegree (−r, r − 1).
The paper unfolds as follows. In the next section we define bundles of coloured posets, describe the associated total coloured poset, and give a number of examples. We also recollect facts from [2] about coloured posets and their homology. In §2 we start with a bundle of coloured posets and introduce a certain bi-complex which (by standard methods) gives rise to a spectral sequence converging to the homology of the total complex. Our task later will be to recognize this as the homology of the total coloured poset of the bundle we started with. In fact we can only do this for a restricted class of bases, the specially admissible ones, and we introduce these in §3. We give a number of examples, including the Boolean lattices. Following this, §4 introduces the technical tools needed to prove the main theorem, namely certain long exact sequences. In §5 we state and prove the main theorem. Finally in §6 we turn to Khovanov homology and construct a new spectral sequence which for a given link has a potentially computable E 2 -page, and which converges to the Khovanov homology of the link.

Bundles of coloured posets
Recall from [2] that a coloured poset is a pair (P, F) consisting of a poset P having a unique maximal element 1 P , and a covariant functor F : P → Mod R , called the colouring. This may be regarded as a representation of P or as a sheaf of modules over P according to taste. A morphism of coloured posets (P 1 , ). This pair must satisfy (i) f (x) = 1 P 2 if and only if x = 1 P 1 , and (ii) for all x ≤ y in P 1 , the following diagram commutes F 2 (f (y)) τx τy Coloured posets and morphisms between them form the category CP R . We also recall from [2] our poset convention that if P has a unique minimal element 0 p , then 0 p < 1 p . In particular the poset with a single element has a 1 but not a 0. If x < y in P and there is no z with x < z < y, then one says that y covers x and writes x < c y.
Throughout this paper, if x ∈ P we write P (x) for the interval P (x) := {y ∈ P | y ≤ x} and P (x) for it's complement P (x) := {y ∈ P | y ≤ x}.
From now on we will just say "poset with 1" to refer to a poset with unique maximal element. The idea of a bundle is that it consists of a collection of coloured posets (the fibres) parametrized by another poset (the base) that encodes instructions for glueing the fibres together.
Thus, to each x ∈ B there is an associated coloured poset ξ(x) = (E x , F x ), the fibre over x. Whenever x ≤ z there is an associated morphism of coloured posets ξ(x ≤ z) = (f z x , τ z x ) from ξ(x) to ξ(z). There is a projection map of (uncoloured) posets π : E → B defined by π(y) = x if and only if y ∈ P x . Definition 1 is akin to the interpretation of a vector bundle (with connection) as a functor from the path space of the base to the category of vector spaces. What is missing from this definition is the analogue of the total space, which is somewhat hidden but which can be constructed as follows.
Definition 2. Let B be a poset with 1 and let ξ : B → CP R be a bundle of coloured posets. The associated total coloured poset is the coloured poset (E, F) defined by: -As a set E = x∈B E x , the union of the fibres, and the partial order on E is defined as follows: -if y, y ′ ∈ E x , so y and y ′ are in the same fibre, then on a morphism y ≤ y ′ as follows: In the last line of the definition we are using the morphism of coloured posets ξ( Proof. It is easily checked that E is a poset. For the colouring we must verify that composition behaves as it should. There are four cases to consider of which we treat one in detail, the others being much simpler. Suppose that y ≤ y ′ and y ′ ≤ y ′′ . We must show that F(y ′ ≤ y ′′ ) • F(y ≤ y ′ ) = F(y ≤ y ′′ ). Consider the following diagram The triangle at bottom commutes since ξ is a functor, the triangle at the right commutes because F x ′′ is a functor and the square commutes by the naturality of τ x ′′ x ′ . Thus the entire diagram commutes. Going up the steps gives F(y ′ ≤ y ′′ ) • F(y ≤ y ′ ) while following the two arrows with bends gives F(y ≤ y ′′ ).
⊓ ⊔ Example 1. Figure 1 gives an example of a bundle over a Boolean lattice of rank two, with the colouring left off.
If B is a poset with 1, and ξ is a bundle over B, then any sub-poset A ⊆ B having a unique maximal element, has an associated bundle obtained by restricting ξ. This is given by the composition of functors A → B → CP R and will be denoted ξ| A . Note that the maximal element of A need not be the maximal element in B. Two instances of particular importance arise when ξ is a bundle over B with total coloured poset (E, F), and for x ∈ B we consider the subposets the interval B(x) and its complement B(x). The interval B(x) has maximal element x and the complement has maximal element the 1 of B. We will denote the total coloured posets of the restricted bundles ξ| B(x) and ξ| B(x) by (E(x), F) and (E(x), F).
Bundles of coloured posets form a category Bund R as we now describe. Definition 3. Let ξ : B → CP R and ξ ′ : B ′ → CP R be bundles of coloured posets over B and B ′ respectively. A morphism (g, η) : ξ → ξ ′ is a map of (uncoloured) posets g : B → B ′ satisfying g(x) = 1 iff x = 1, together with a natural transformation η from ξ to ξ ′ • g.
By fixing the base poset B and restricting morphisms to those with g = id, one obtains a subcategory of bundles over B denoted Bund R (B).
Associating the total coloured poset to a bundle is functorial: given a bundle ξ : B → CP R let E(ξ) = (E, F) be the associated total coloured poset. For a morphism (g, η) : ξ → ξ ′ between bundles ξ : B → CP R and ξ ′ : B ′ → CP R , we define a morphism of coloured posets E(g, η) : E(ξ) → E(ξ ′ ) in the following way. Recall that we must define a poset map f : E → E ′ and a collection {σ y : F(y) → F ′ (f (y))}, and that for x ∈ B, the natural transformation η gives a morphism of coloured posets, courtesy of the natural transformation η, from which one can deduce that (f, σ) is a morphism of coloured posets. Moreover, it preserves fibres. Thus given a morphism (g, η) : ξ → ξ ′ we have assigned a morphism of coloured posets E(g, η) = (f, σ). Since composition behaves well we have, There is a homology of coloured posets and our main interest in this paper is to compute it for the total coloured poset of a given bundle. We now briefly recall the relevant definitions-for more details see [2]. Let (P, F) be a coloured poset and define a chain complex S * (P, F) by letting S k (P, F) = 0 for k < 0; for k > 0 let the direct sum over all multi-sequences x 1 ≤ x 2 ≤ · · · ≤ x k ∈ P \ 1, and S 0 (P, F) = F(1). The differential d k : The homology H * (P, F) of (P, F) is then the homology of this complex, and this gives a functor H * : CP R → GrMod R .
When working with homology it can be convenient to use a smaller complex than the one above. One defines C * (P, F) identically to S * (P, F), but with the additional requirement that the indexing x 1 x 2 · · · x k appearing above are now sequences rather than multi-sequences, i.e. there are no repeats. Clearly C k ⊂ S k , and the differential is taken to be the restriction to C k of the differential on S k (so that C * is a subcomplex of S * ). One can show that there is a homotopy equivalence of chain complexes C * (P, F) ≃ S * (P, F) as in [2, §2].
Returning to the situation of a bundle ξ : B → CP R , these constructions give important colourings of the base. Let E(ξ) = (E, F). For each positive integer q ≥ 0 we have the q-chain colouring We can also colour the base using the homology of the fibres. Let ξ : This defines a colouring on B and so a coloured poset (B, H fib q (ξ)). If (g, η) : ξ → ξ ′ is a bundle morphism and x ∈ B, then η(x) : ξ(x) → ξ ′ g(x) induces a map in homology . Thus by taking homology of fibres we have a functor Bund R → CP R .

A bicomplex and its total complex
Our main theorem is a Leray-Serre type spectral sequence for bundles. Fundamental to its construction is a certain bicomplex which we now describe. Let ξ : B → CP R be a bundle with total coloured poset E(ξ) = (E, F). From the previous section we have an infinite family of coloured posets (B, S q ), where the base has been q-chain coloured for each q ≥ 0. Now set Explicitly, C p (B, S q ) is a direct sum of modules in the colouring of B indexed by length p sequences in B \ 1, so a typical element is a sum of µx 1 . . . x p with the x 1 < · · · < x p ∈ B \ 1 and µ ∈ S q (E x 1 , F x 1 ). Thus µ is in turn a sum over λy 1 . . . y q with the y 1 ≤ · · · ≤ y q ∈ E x 1 \ 1 x 1 and λ ∈ F x 1 (y 1 ). We will thus write is the induced coloured poset morphism. Fixing p we have K p, * a complex under the differential d v : It is tedious but straightforward to check that There is then an associated total complex (T * , d) where From the general construction of a spectral sequence from a bi-complex we have, Proposition 2. Let ξ be a bundle of coloured posets with base B and total poset (E, F). Then there is a spectral sequence Example 6. Let B be a poset with a unique minimal element 0 satisfying 0 < 1, and let (P, F) be a coloured poset. Then T * (B × (P, F)) is acyclic. To see this notice that (B, H fib q (ξ)) is a coloured poset with a constant colouring (by H q (P, F)) and thus, since B has a 0, gives [2, Example 8] that the E 2 -page of the above spectral sequence is trivial.

Admissible and specially admissible posets
where ξ is a bundle over a certain class of posets that we now introduce. Let P be a poset with 1, and for x ∈ P recall from §1 the definition of the interval P (x) and its complement P (x).

Definition 4. A poset P with 1 is admissible if and only if there exists an
On the far left of Figure 2 we have an artificial example of a poset, admissible courtesy of the x shown. In general, L(y) has unique maximal element the 1 of P with 0 L(y) < 1 by our standing poset convention, so L(y) necessarily contains at least two elements if P is to be admissible. In particular the poset n = 1 < 2 < · · · < n, with Hasse diagram on the far right of Figure 2 is non-admissible when n ≥ 3.
Example 7. For B(X) the Boolean lattice of all subsets of X ordered by inclusion, |X| is the rank of B(X). Then B(X) is admissible for X = ∅. This is vacuous in rank 1 (where B(X) is isomorphic to the poset 2 = 1 < 2). If |X| > 1 then any subset A ∈ B(X) covered by X is of the form X \ x for some x ∈ X, and the interval (also a Boolean lattice) B(A) consists of subsets of X not containing x. For Y ∈ B(A) \ {A} the set L(Y ) consists of subsets of X containing the element x and the elements of Y . Thus 0 L(Y ) = Y ∪ {x} is a unique minimal element for L(Y ). The second example of Figure 2 is Boolean of rank 3.
Example 8. Let (W, S) be a Coxeter group with length function ℓ : W → Z ≥0 and reflections T (see [3] for this and other basic facts about Coxeter groups). Write The result is a graded (by ℓ) poset with 0 (the identity of W ) called the Bruhat (-Chevelley) poset of W . A simple example is the Bruhat poset of the Weyl group of type A n 1 , which is the Boolean lattice of rank n. Two warnings: w ′ → w is not the same as w ′ < c w, and (W, ≤) is not in general a lattice.
If W is finite then the Bruhat poset also has a 1: the element of longest length in W . In fact, the Bruhat poset of any finite Coxeter group is admissible, and this and other aspects will be investigated in a forthcoming paper. For now, we leave it to the reader to check that the Bruhat poset for the symmetric group S 4 (the Weyl group of type A 3 ), which is the third example in Figure 2, is admissible for any x < c 1.
Suppose now that ξ : B → CP R is a bundle with admissible base B and (E, F) = E(ξ). The admissibility of the base has the following useful consequence for E. Let x ∈ B, with interval B(x), complement B(x), and restricted bundles (see Proof. We have π(y) ∈ B(x) and by admissibility, the subposet L(π(y)) has a unique minimal element z 0 ∈ B(x). The bundle furnishes us with a morphism of coloured posets (E π(y) , F π(y) ) → (E z 0 , F z 0 ), which is comprised in particular of a poset map f z 0 π(y) : E π(y) → E z 0 . We leave it to the reader to verify that the minimal element we seek is f z 0 π(y) (y) ∈ E z 0 . ⊓ ⊔ Admissible posets are an important intermediary concept: the bundle results of §4 are for instance true when the base is admissible. For the main theorem of §5 we require something stronger: that the base poset B is admissible for some x < c 1, and when split into the associated interval B(x) and its complement B(x), these two are also admissible, and when these are split the results are admissible, and so on. Thus, we will employ an inductive approach, resulting in a collection of posets scattered on the workbench, and we will need each piece to be admissible. The definition is best formulated recursively: Definition 5. A poset P with 1 is specially admissible if and only if either 1. P is Boolean of rank 1, or 2. P is admissible for some x < c 1 with P (x) and P (x) specially admissible.
Example 9. We have already seen that the Boolean lattice of rank n > 0 is admissible (for any x < c 1) with the P (x) and P (x) Boolean of rank n − 1. The Boolean lattice of rank 1 is specially admissible by definition, so Boolean lattices of any rank are specially admissible by induction.
Example 10. Another simple family of specially admissible posets are the Bruhat posets of type I 2 (m), m ≥ 2, which have Hasse diagrams (laid out sideways with order running from left to right) as in Figure 3. For any x < c 1, the interval P (x) is the Bruhat poset of type I 2 (m − 1), and its complement P (x) is Boolean of rank 1. As any two elements in I 2 (m) are comparable, we have that for y ∈ P (x) \ x, the poset L(y) is also Boolean of rank 1. Thus I 2 (m) is admissible. The Bruhat poset of type I 2 (2) is the Boolean lattice of rank 2, and so by induction, I 2 (m) is specially admissible.
Example 11. Looking back at Figure 2, one can see (after quite an inspection) that the Bruhat poset A 3 is specially admissible.

Some exact sequences
This section contains the technical tools needed for the proof of the main theorem. As we will see in §5, the key step is the identification of the homology of T * (E, F) with the coloured poset homology H * (E, F) in the case where the base poset is specially admissible. To achieve this we will use an inductive argument, splitting the base into two pieces and then using the two long exact sequences presented below in Propositions 3 and 4.
Throughout this section let B be a poset, admissible courtesy of some x < c 1 B . Let ξ be a bundle of coloured posets with base B and E(ξ) = (E, F). Let B(x) be the interval from §1 and B(x) its complement, with the associated restricted bundles having total posets (E(x), F) and (E(x), F). Proof. It is immediate from the definition of the differential that T * (E(x), F) is a subcomplex of T * (E, F) and thus for some quotient Q ′ * there is a short exact sequence Explicitly, Q ′ * can be described as the sum over xzy = x 1 . . . x i z 1 . . . z j y 1 . . . y k with i > 0 and i + j + k = n, and where x 1 . . . x i is a sequence in B(x), z 1 . . . z j a sequence in B(x) \ 1, and y 1 . . . y k a multi-sequence in E x 1 \ 1 Ex 1 .
Here, as elsewhere E x 1 is the fibre over x 1 . Notice that the condition i > 0 means that the sequence x 1 . . . x i z 1 . . . z j must start in B(x). When i > 1 the differential in Q ′ * sends λxyz to, and for λxz 1 . . . z j y 1 . . . y k the image is Now let D ′ * be the submodule of Q ′ * consisting of the modules indexed by the xzy = x 1 . . . x i z 1 . . . z j y 1 . . . y k with x i = x. In other words, the last element of the sequence in B(x) is not the maximum element x of this interval. It is immediate that D ′ * is a subcomplex of Q ′ * and thus for the quotient A ′ * there is a short exact sequence We show that D ′ * is acyclic, and this allows us to finish the proof easily: one sees that given by λx 1 . . . x i−1 xy 1 . . . y k → λx 1 . . . x i−1 y 1 . . . y k is an isomorphism, and so using the long exact sequence associated to (3), we have H n (Q ′ * ) ∼ = H n−1 (T * (E(x), F)). Plugging this into the long exact sequence associated to (2) gives the result.
The remainder of the proof is thus devoted to showing that D ′ * is acyclic. We filter D ′ * as follows: where the sum is over the xzy = x 1 . . . x i z 1 . . . z j y 1 . . . y k satisfying 1 ≤ i ≤ s. There is an associated spectral sequence converging to H * (D ′ * ) having where the sum is over the xzy = x 1 . . . x s z 1 . . . z j y 1 . . . y k with j +k = t. The differential d 0 sends s, * to be the submodule of elements of the form λx 1 . . . x s z 1 . . . z j y 1 . . . y k , where the sequence x 1 . . . x s is the given one x. Endowing this with the differential gives a subcomplex.
Let L(x s ) be the subposet of B with elements {z ∈ B(x) | x s ≤ z} as in Definition 4, and having a unique minimal element courtesy of the admissibility of B. By direct comparison of the definitions on both sides we see that there is an isomorphism of complexes and by Example 6, the right hand side (and hence U x * ), is acyclic. We now show that any cycle in E 0 s, * is a boundary. A typical cycle has the form σ = λ xzy xzy and can be decomposed as where σ x = x ′ =x λ x ′ zy x ′ zy. Now, if xzy and x ′ z ′ y ′ are two sequences occurring in σ and x = x ′ then d 0 (xzy) and d 0 (x ′ z ′ y ′ ) are in disjoint summands of d(σ). Thus d 0 (σ) = 0 implies that d 0 (σ x ) = 0 for all σ x . In other words, as σ is a cycle, each σ x is a cycle. Clearly we have σ x ∈ U x * and hence since U x * is acyclic, there exists τ x such that d 0 (τ x ) = σ x . Thus d 0 ( τ x ) = σ x = σ and σ is a boundary as claimed.
Thus, for each s the complex E 0 s, * is acyclic and hence the E 1 -page of the spectral sequence is trivial, showing that D ′ * is acyclic as required.

⊓ ⊔
In the proof of the main theorem in §5, we will find it more convenient to use We leave it to the reader to check that this is indeed an isomorphism.
There is a similar long exact sequence for the homology of the total coloured poset (E, F). It is a generalization of [2, Theorem 1] from bases that are Boolean of rank 1 to bundles over admissible bases. The proof is very similar to the last proposition so we will be briefer. and as above we can describe Q * explicitly as where xz = x 1 . . . x i z 1 . . . z j with i > 0, i + j = n and x 1 . . . x i a multi-sequence in E(x) and z 1 . . . z j a multi-sequence in E(x) \ 1 E . Now let D * be the subcomplex of Q * consisting of the modules indexed by the xz = x 1 . . . x i z 1 . . . z j such that x i = 1 E(x) , and A * the quotient of Q * by D * with quotient map q. We show, as above, that D * is acyclic, so that if where the sum is over the xz = x 1 . . .
where the sum is over xz = x 1 . . . x s z 1 . . . z t . The differential d 0 is as follows (where the overall sign is given by the parity of s):

A Leray-Serre spectral sequence
Here is the main theorem of the paper.
In §2, Proposition 2, we constructed a spectral sequence F)). To prove the theorem it therefore suffices then to find a quasi-isomorphism φ : T * (E, F) → S * (E, F).
We begin by defining a chain map φ : T * (E, F) → S * (E, F). Let ξ : B → CP R be a bundle, with B any base, and (E, F) the total coloured poset. Let (x, y) be a pair consisting of a sequence x = x 1 < · · · < x n in B and a multi-sequence y = y 1 ≤ · · · ≤ y m in E x 1 . Note that we allow 1 ∈ B to be an element of x and 1 x 1 ∈ E x 1 to be an element of y. For each 1 ≤ i ≤ n, the bundle furnishes us with a poset map f x i a multi-sequence in E x i (with y 1j = y j ). The resulting m × n collection {y ij } may be placed on the vertices of a rectangular array and we join these by placing oriented edges between y i,j and y i,j+1 and between y i,j to y i+1,j . We refer to this as the grid associated to the pair (x, y). We have comparability of elements in the columns of the grid, y ij ≤ y i,j+1 , thanks to (5), and also along the rows, y ij < y i+1,j , via the definition of the ordering on E. Given x 1 < · · · < x p ∈ B \ 1 and y 1 ≤ · · · ≤ y q ∈ E x 1 \ 1 x 1 with p, q > 0, we consider the grid associated to (x = x 1 < · · · < x p < 1, y = y 1 ≤ · · · ≤ y q < 1 x 1 ). An (x, y)-multi-sequence is a multi-sequence in E of the form, with the z k elements of the grid satisfying z 1 = y 11 , z p+q = y p+1,q or z p+q = y p,q+1 , and for any 1 ≤ k ≤ p + q, if z k = y ij , then z k+1 ∈ {y i+1,j , y i,j+1 }. In other words an (x, y)-multisequence consists of the ordered collection of vertices in an oriented path from y 11 to either y p+1,q or y p,q+1 . Figure 4 exhibits a grid and two sample paths.
Such a path partitions the grid into two halves-the upper-left half and the lower-right half. Given an (x, y)-multisequence z we define m(z) to be the number of squares of the grid in the lower-right half.

Proposition 5. φ is a chain map, and if
Proof. The inclusion of complexes is clear, so that it remains to show that φd T (λxy) = d S φ(λxy) for any λxy ∈ T p+q (E, F). There are two parts to the proof: first we show that every term of φd T also occurs as a term of d S φ, and then that any extra terms in d S φ occur in ± pairs, and hence cancel.
Ignoring signs for the moment, the terms of φd T are precisely the terms of d S φ that are obtained by taking a z = z 1 . . . z p+q of φ(λxy) and letting the differential d S omit z i as, or by letting d S omit the first element z 1 or the last element z p+q . In words, if d S omits z i , then z i−1 , z i+1 are both in the same row, or both in the same column, as z i . Indeed, the situation above left arises in φd T when d T omits some x ∈ B that indexes the column in which z i lies, and then mapping to the multi-sequenceẑ = z 1 . . . z i . . . z p+q corresponding to a path in the grid in which the i-th column of the original grid has been deleted. The situation above right is similar, with an appropriate y ∈ E x 1 omitted by d T this time. Chasing definitions one can verify that the coefficients of such terms agree up to sign.
With that out of the way, the terms of d S φ that don't arise at all in φd T are those where d S omits from a multi-sequence z a z i sitting on a "corner": Now to the signage, where we check first that these extra terms occur in ± pairs and so cancel. Suppose we have such a term, arising when d S omits a corner from a z ∈ φ(λxy). There is a unique repetition of this term, arising by taking the z ′ ∈ φ(λxy) that travels the other way around this square and letting d S omit the other corner (z and z ′ thus differ only in the segment shown in the figure above). It is easy to check that z, z ′ acquire different signs from φ and that d S preserves this difference. Thus, the extra terms in d S φ can be coupled into ± pairs as required. Now to the terms common to φd T and d S φ, where we show that they occur with the same signs. Let λxy = λx 1 · · · x p y 1 · · · y q and apply φ to give a sum of (x, y)-multi-sequences with the multi-sequence z having sign (−1) α(q)+m(z) . Now applying d S to this we get a sum of terms of the form z 1 . . .ẑ i . . . z p+q picking up a sign −(−1) i . Thus the total sign of the summand of d S φ(λxy) indexed by Now there are two cases to consider, where the first is if z i−1 , z i and z i+1 are all in the same column (see Figure 5). Suppose the column in question is that of x k for some k. This implies that the row corresponding to z i is that indexed by y i−k+1 . Thus in order to obtain via φd T the summand indexed by z 1 . . .ẑ i . . . z p+q we must as a first step consider the summand of d T (λxy) indexed by the term x 1 · · · x p y 1 · · ·ŷ i−k+1 · · · y q . This picks up the sign −(−1) p+q+(i−k+1) . After applying φ the summand indexed byẑ = z 1 . . .ẑ i . . . z p+q picks up an additional (−1) α(q−1)+m(ẑ) so the total sign is − (−1) p+q+(i−k+1)+α(q−1)+m(ẑ) .
Now,ẑ lives in a new grid obtained from the original one by deleting the (i − k + 1)-th row. Thus from which we deduce By checking the various possibilities for α(q) (or by observing that q + [q − 1] = [q]), we see that the signs of (7) and (8) coincide. The second case, where z i−1 , z i and z i+1 are all in the same row, is entirely similar.

⊓ ⊔
As a matter of interest we note that φ is the unique chain map of this form. Thinking degree by degree, if φ is to be a chain map, then its definition at T p+q is determined by its definition at T p−1,q and T p,q−1 , as well as the differentials d T and d S . Thus, once we have defined φ at T 0+0 , we have no choice for the definition at T 0+1 , T 1+0 , and these in turn determine the definition at T 1+1 , and so on, working our way upwards and rightwards from the bottom lefthand corner of the bi-complex. As T and S coincide in degree 0, the simplest choice for φ is the identity, and our map is precisely the chain map determined by this choice.
Proposition 6. If B is finite and specially admissible then φ is a quasi-isomorphism.
Proof. Let B be admissible via x < c 1. Recalling the notation used in §4 there is a morphism of short exact sequences: where φ ′ is the map induced on the quotients by the second part of Proposition 5. It is easy to check that the diagram commutes. Using the functorality of the long exact sequence in homology, we have the following commutative diagram We leave it to the reader to check that the diagram commutes, with ψ := πq and ψ ′ := (−1) j π ′ q ′ the quasi-isomorphisms of §4. In particular, we have φ ′ * = ψ −1 * φ * ψ ′ * . We now argue by induction on the cardinality of the base. If B is Boolean of rank one with two elements 0 < 1, then B is admissible for x = 0, E(x) is the fibre (E 0 , F 0 ) and E(x) is the fibre (E 1 , F 1 ). Both of these are the total spaces of a bundle over the trivial poset with a single element. For such a bundle T * = S * and φ = id, inducing an isomorphism H * T * → H * S * . Thus, we have the result for B using the diagrams above and the 5-lemma.
In general, as B is specially admissible, so are B(x) and B(x), and so the proof is again finished by induction and the 5-lemma.
We remark here that our proof relies heavily on having a specially admissible base because in general the long exact sequences used in the previous proposition do not exist. Whether the theorem remains true in greater generality remains an open question.
Consider the crossings of D ′ outside the disk. We can decompose (B N +1 , F D ′ ) as a bundle over the Boolean lattice B N on these crossings: the fibres correspond to the two possible resolutions of the crossing in the disk as in Figure 6 (with n = 2).
In particular, the complex C * for the fibres is id ⊗i ⊗m⊗id ⊗j : . Now apply the main theorem of [2], turning this isomorphism into the required one between the Khovanov homologies.

An application to knot homology
In this section we give an application of the spectral sequence of Theorem 1 to knot homology. We will write the results in terms of Khovanov homology, but in fact one can be more general-see the remarks at the end of the section. The idea is the following. Starting with a link diagram we choose a subset of the crossings to be fixed and form a cube of link diagrams by resolving the remaining crossings in the way familiar in Khovanov homology (see Khovanov [4] and Bar-Natan [1] ). We are now at liberty to take the Khovanov homology of each link diagram at a cube vertex and moreover to consider the induced maps associated to Morse moves along the edges. From this we can form a complexà la Khovanov. The main result is that there is a spectral sequence converging to the Khovanov homology of the original diagram with the homology of this new complex at the E 2 -page.
Our conventions for shifting complexes will be that for a tri-graded vector space W * , * , * we define W * , * , * [a, b, c] by (W * , * , * [a, b, c]) i,j,k = W i−a,j−b,k−c . Similar conventions apply to bi-graded and singly graded vector spaces.
First we outline our grading and normalisation conventions in constructing Khovanov homology. Let D be a link diagram with N crossings. The unnormalised Khovanov homology of the diagram where V is the usual graded vector space defining Khovanov homology, k α is the number of circles appearing in smoothing α and rk is the rank function on the Boolean lattice. The differential d : If the diagram D is oriented, with N + positive and N − negative crossings, we can also define the normalised Khovanov homology to be, The reader should note that this differs from the original conventions followed by Khovanov [4] and Bar-Natan [1], where unnormalised homology is defined via a cochain complex C * , * (D) which is normalised to define KH * , * (D), the latter being an invariant of oriented links. The conventions we use are related to the original ones by Now to the complex referred to above. Let D be an oriented link diagram with N crossings. Choose k crossings c 1 , . . . , c k and call them the fixed crossings. The remaining ℓ = N − k crossings are the free crossings, and we assume these too have been ordered.
Resolving the free crossings in one of the two familiar ways yields 2 ℓ diagrams which we place at the vertices of the Boolean lattice B ℓ (a cube in more usual Khovanovology). Figure 7 illustrates this for the knot 7 5 with the four fixed crossings as shown in red.
For x ∈ B ℓ let D x denote the corresponding link diagram. Note that D x does not in general inherit an orientation from D, and we will be treating D x as an unoriented diagram.
We now associate to each vertex x of the cube the bigraded module the unnormalised Khovanov homology of the (unoriented) diagram D x shifted in the second degree by the rank of x in B ℓ . If x < x ′ ∈ B ℓ then the diagrams D x and D x ′ are identical except in a small disc within which one of the zero smoothings in D x turns in to a 1-smoothing in D x ′ . By using the multiplication or comultiplication in the familiar way (geometrically using the saddle cobordism) there is a chain map, This map has bi-degree (0, −1). Thus, after taking homology and shifting, we have a homomorphism of bidegree (0, 0), We now define the complex K * , * , * (D; c 1 , . . . , c k ) by setting The differential is defined using the maps δ x ′ x in the usual way (and using the usual signage conventions in Khovanov homology). It has tri-degree (1, 0, 0), and is indeed a differential because all squares in the cube anti-commute (even though these are now Morse-move frames in a movie of link diagrams) because saddles can be re-arranged without any unwanted side effects.
It is important to note that KH * , * , * (D; c 1 , . . . , c k ) depends on the set of fixed crossings and is with respect to a specific diagram. However, KH * , * , * (D; c 1 , . . . , c k ) is invariant under re-ordering of the fixed and free crossings. If the set of crossings is empty then we recover the unormalised Khovanov homology.
With these preliminaries let us now state the theorem. The r-th differential in the spectral sequence has bidegree (−r, r − 1).
Proof. The procedure for constructing KH * , * (D) includes (in our language) the Khovanov colouring F of the Boolean lattice B k+ℓ , on the set of crossings of the diagram. It is a graded colouring in that it takes values in graded modules. We have been given a subset of the crossings in the form of the fixed crossings. By Example 4 we can decompose (B k+ℓ , F) as a bundle ξ over the Boolean lattice on the free crossings, i.e. a copy of B ℓ . Note that the total space of this bundle is (B k+ℓ , F).
Fixing i and applying Theorem 1 we thus have a spectral sequence where the colouring F is the restriction of the Khovanov colouring to grading i. Recall from [2] that KH * , * (D) ∼ = H * , * (B k+ℓ , F), Thus the spectral sequence above converges as required to KH * ,i (D), and it remains to identify the E 2 -page. It is clear that for x ∈ B ℓ the fibre over x is isomorphic to B k equipped with the Khovanov colouring F x of the diagram D x shifted by rk(x). Thus the homology of the fibre over x is given by H * , * (B k , F x )[0, rk(x)]. Using again the isomorphism between Khovanov homology and coloured poset homology, we see that the homology of the fibre is isomorphic to KH * , * (D x )[0, rk(x)] = V * , * (x). The induced maps between fibres are by definition the maps δ x ′ x . Thus we have coloured B ℓ by the modules V * , * (x) and the maps δ x ′ x . The E 2 -page of the spectral sequence is the coloured poset homology of this coloured poset and KH * , * ,i (D; c 1 , . . . , c k ) is the Khovanov homology of it. Appealing again to [2], these two homologies agree, and hence the result is proved.
⊓ ⊔ Remark 1. In practice it may well be useful to apply Reidemeister moves to D x -a quick look at Figure 7 should be enough to see why. Since we are using unnormalised Khovanov homology to colour fibres we must be a little careful as shifts need to be introduced. The best way to proceed is to choose an orientation for D x and count the number of positive and negative crossings n + and n − . One can then express V * , * (x) in terms of normalised Khovanov homology as V * , * (x) = KH * , * (D x )[n + , 2n − − n + + rk(x)], and now of course we are free to compute KH * , * (D x ) by applying Reidemeister moves if necessary.
Remark 2. All of the above works in more generality than is stated. Indeed one can do a similar thing given a link homology theory which can be extended to knotted resolutions. An important example of this is Khovanov-Rozansky theory which is constructed from a cube of resolutions where each resolution can be viewed as a 4-valent graph. By fixing some subset of crossings we get a (smaller) cube of knotted 4-valent graphs. Khovanov-Rozansky homology has been extended to such knotted graphs by Wagner in [5]. By following through the procedure above one then gets a spectral sequence converging to Khovanov-Rozansky homology with a diagram-dependent, but in principle calculable, E 2 -page.