A sheaf-theoretic description of Khovanov s knot homology

Transcription

1 Journal of Knot Theory and Its Ramifications c World Scientific Publishing Company A sheaf-theoretic description of Khovanov s knot homology WILLIAM D. GILLAM Department of Mathematics, Columbia University, New York, NY ABSTRACT We give a description of Khovanov s knot homology theory in the language of sheaves. To do this, we identify two cohomology theories associated to a commutative diagram of abelian groups indexed by elements of the cube {0, 1} n. The first is obtained by taking the cohomology groups of the chain complex constructed by summing along the diagonals of the cube and inserting signs to force d 2 = 0. The second is obtained by regarding the commutative diagram as a sheaf on the cube (in the order-filter topology) and considering sheaf cohomology with supports. Included is a general study of sheaves on finite posets, and a review of some basic properties of knot homology in the language of sheaves. Keywords: Khovanov knot homology, sheaf, finite poset Mathematics Subject Classification 2000: 57M25, 57M27, 55N30, 55N35 1. Sheaves on finite spaces A topological space X determines a partially ordered set (poset) with the same underlying set X by declaring x y iff x is in the closure {y} of {y} (i.e. x is a specialization of y). A continuous map f : X Y satisfies x {y} = f(x) {f(y)}, hence induces a morphism between the corresponding posets. Similarly, any poset P = (P, ) can be regarded as a topological space whose basic open sets are the sets of the form U x := {y P : x y}, where x P. A general open set in this topology is an order-filter: a subset U P such that, whenever x U and x y, y U. The topology of a finite topological space is determined by specialization (a set is closed iff it is closed under specialization), hence the above associations are easily seen to yield an equivalence between the category of finite topological spaces and the category of finite posets. We suppress notation for this equivalence, regarding a poset as a topological space (and vice-versa). For simplicity, we assume our spaces (and posets) are sober in the sense that x y and y x = x = y. 1

2 2 William D. Gillam This is harmless since we can always quotient by the corresponding equivalence relation to obtain a sober space; the category of sheaves (our ultimate object of interest) is unchanged by this sobrification process Sheaves We will be interested in sheaves on finite topological spaces, which are also studied in [1,3,4,14]. a We will use the language of posets, since it is most natural in the applications we have in mind. For a poset P, and an element x P, the basic open set U x is the smallest open set containing x, so the stalk F x of a sheaf F on (P, ) at x is given by F (U x ), thus U x is good in the sense that the higher derived functors of the (usual) global section functor vanish for any sheaf on U x. These open sets are also the gross objects of the topology on (P, ) (i.e. they have no non-trivial covers). The basic closed sets Z x := {y P : y x} are exactly the irreducible closed sets (in the sense of noetherian topology). The intersection of two good open sets U x, U y is good iff x and y have a least upper bound in P so existence of good covers is related to lattice properties. Now suppose that F : x F (x) is a covariant functor from P (regarded as a category where there is a unique morphism x y iff x y) to an abelian category A. We may then regard F as a sheaf on P by declaring { } F (U) := (a x ) x U F (x) : F xz (a x ) = F yz (ay) x, y z Here we are abusing notation by assuming A = Ab is the category of abelian groups since we really mean F (U) := lim F (x), where the inverse limit is taken over all maps F xy for x y, x, y U. Clearly this is a sheaf whose stalk F x is just F (x) A. For x y in P, the corresponding map F xy : F (x) F (y) is recovered from the sheaf F as the map on stalks F x F y induced by the specialization x {y}. From this, one sees easily that every sheaf on P arises from a unique functor P A in the above manner, so in what follows we will identify sheaves on P with functors from P to A. This is also a consequence of the general fact (c.f. [4]) that if C is a finite site with at most one morphism between any two objects then the inclusion of the full subcategory C 0 C of gross objects (with the trivial Grothendieck topology) induces an equivalence of topoi. a It is interesting to read the older references which were written before [7] became common knowledge. Many results proved in the pre-1970 papers are trivial consequences of general facts about sheaves of abelian groups on topological spaces established in [7].

3 August 4, :17 WSPC/INSTRUCTION FILE ksheaf-ws A sheaf-theoretic description of Khovanov s knot homology 3 between the site of sheaves on C 0 (which are just presheaves on C 0 ) and the site of sheaves on C. A morphism of sheaves F G is a natural transformation of functors. A sequence of sheaves is exact iff 0 F F F 0 0 F (x) F (x) F (x) 0 is exact for all x P. The kernel and cokernel sheaves are formed in the obvious way. Denote the category of sheaves on P with values in A by A(P ) Resolutions If j x : {x} P is the inclusion of a point and A A then, regarding A as a sheaf on {x}, the sky-scraper sheaf j x A on (P, ) is given by setting { A, y x (j x A)(y) := 0, otherwise The restriction maps are Id A when y z x and 0 otherwise. These sky-scraper sheaves are flasque and if I is an injective object of A then the sheaf j x I is an injective object of A(P ). Indeed, this is the standard injective object used throughout [7]: one easily checks that to give a morphism of sheaves F j x I is to give a morphism on the stalk b F x I so that Hom A(P ) (, j x I) is the composition of the exact stalk functor F F x and the exact functor Hom A (, I). Following [14], we can check directly that j x I is injective (this will be useful below). Given a solid diagram F G F (x) G (x) f j x I in A(P ) as on the left with F G monic, the map F (x) G (x) in the right solid diagram is also monic in A, so there is a lift f(x) : F (x) I as indicated. Declaring { f(x)fxy, y x f(y) := 0, otherwise f(x) defines a morphism f : F j x I making the left diagram commute. Define the rank of P to be rk(p ) := max{t N : x 0 < < x t in P }. I b since Hom(F, j G ) = Hom(jx 1 F, G ) by adjointness

4 4 William D. Gillam The rank is also the noetherian dimension, thus one can see that noetherian dimension is monotone (does not increase) for subspaces of posets. This is not true of arbitrary noetherian spaces. The injective dimension i. dim(a) of an object A in an abelian category A is the length of the shortest finite injective resolution if one exists, otherwise; an injective object has i. dim = 0 and 0 has i. dim = 1. The injective dimension i. dim(a) is the supremum over objects (if it exists). Say A satisfies (*) if for any diagram D g 0 A f B C 0 in A with B injective, with exact row, and with i. dim(c) n, we have i. dim(cok fg) n. Every vector space is injective, so (*) holds trivially there, and (*) holds in Ab because a quotient of a divisible abelian group is divisible (the only interesting case of (*) for Ab is n = 0 since i. dim(ab) = 1). The following theorem slightly generalizes [14, Prop. 2.7]. We only bother to state and prove it because we want to explain how to find short injective and flasque resolutions in Ab(P ). Theorem 1.1. If A satisfies (*), then i. dim(a(p )) i. dim(a) + rk(p ). Proof. There is nothing to prove if i. dim(a) =, so we can assume d := i. dim(a) is finite. For each F A(P ), and each x P, there is a monomorphism d x : F (x) I x with I x injective whose cokernel C x has i. dim(c x ) d 1. The cokernel sheaf of the map F x j x I x =: I 0 given by F (y) I 0 (y) = I z, via d z F yz z y has stalk at y given by z y Cok(d zf yz ). I claim this stalk has injective dimension at most d 1. Indeed, it is a product of objects of A each with injective dimension at most d 1 because we have a diagram F y z y F yz 0 F z I z C z 0 and A satisfies (*). We may repeat this construction to find an exact sequence 0 F I 0 I d 2 I d 1, where the cokernel sheaf G of the last map has injective stalks. Let M 0 be the set of maximal elements of (P, ) and inductively define M i+1 to be the set of maximal elements of P \ M i. Perform the same construction as above to define an injection G J 0, except now we may assume that each map d x is an isomorphism so that

5 A sheaf-theoretic description of Khovanov s knot homology 5 C x = 0. It follows that the cokernel sheaf of this injection has injective stalks and has stalk at x equal to 0 for each x M 0. Doing the same construction on this cokernel sheaf yields an injection whose cokernel stalks are injective and equal to 0 at each x M 0 M 1, and so on, so we may extend our first exact sequence to an exact sequence 0 F I 0 I d 2 I d 1 J 0 J t 0, where t = rk P. The last map is surjective because the stalks of its cokernel vanish at each point of M 0 M t = P. There is a similar class of projective objects in A(P ) (see [14] and below) so that one can easily prove the dual theorem with the assumption on projective dimension dual to ( ) Cohomology with support For x P define a functor Γ x : A(P ) A by ( Γ x (F ) := Ker x<y F xy : F (x) x<y F (y) ) F (x). (1.1) The functor Γ x is the composition of the exact restriction functor F F U x and the global sections with support in Z x functor Γ Zx. The latter functor is left exact, so Γ x is also left exact. Its right derived functors are called the cohomology with support (see [7, Exer. III.2.3], [8, II.9], or [6]) and are denoted H i x := R i Γ x. These derived functors are acyclic for flasque objects so they can be computed algorithmically by taking resolutions by sky-scraper sheaves as in the proof of Theorem Direct and inverse images For a continuous map f : (P, ) (Q, ) and sheaves F A(P ), G A(Q), the push-forward (direct-image) and pull-back (inverse-image) sheaves f F A(Q) and f 1 G A(P ) are given by (f F )(x) = lim {F (y) : y Q, x f(y)} (1.2) (f 1 G )(x) = G (f(x)) where the first limit is the inverse limit over the restriction maps F yz for y z and x f(y). A subset Q P is locally-closed iff U x Q is closed in U x for every x P. Equivalently, Q is locally closed iff, for every x y z in P with z Q, we have y Q. If j : Q P is the inclusion of a locally-closed subset, then j! : A(Q) A(P ) (the pushforward with support c in Q) and its right adjoint c In [7] this is only defined for open sets (which are always locally-closed) and is called extension by zero. We follow the treatment in [8, II.6].

6 6 William D. Gillam j! : A(P ) A(Q) ( j upper shriek ) are given by { F (x), x Q (j! F )(x) = 0, x / Q (j! F )(x) = Ker F xy : F (x) y P \Q, x y y P \Q, x y F (y). (1.3) One can see directly that when Q is closed j = j! and that j 1 = j! when Q is open (c.f. [8, Prop. 6.9]). It is a nice exercise to show that if A A is projective and A is the locally-constant A-sheaf on Z x (for some x P ), then j A = j! A (where j : Z x P is the inclusion) is a projective object of A(P ). In particular, the locally constant Z-sheaf Z in Ab(P ) is projective when P has a maximum element. 2. Sheaves on the cube The set of subsets of a (finite) set S can be regarded as a (finite) poset P(S) under the order by inclusion. We let S[i] be the set of subsets of S of cardinality i (so S[0] = { }). The basic open sets of P(S) satisfy U x U y = U x y Cube complex Given a sheaf F on P(S), we form its cube complex C (F ) by defining C i (F ) := F (x). x S[i] To define the boundary maps for C (F ), we must choose a linear ordering of S. We then set d i : C i (F ) C i+1 (F ) equal to s 0 < <s i j=0 i ( 1) j F (x\{sj })x. Up to isomorphism, the complex C (F ) is independent of the chosen ordering of S (like the ordered Čech complex of a cover). A map F G of sheaves on P(S) gives rise to a map C (F ) C (G ) of complexes. Theorem 2.1. Cohomology with support at ( 1.3) coincides with cohomology of the cube complex: H i (P(S), F ) = h i (C (F )) for every sheaf F on P(S). Proof. Both sides form (additive) covariant -functors A(P ) A (c.f. [5, 2.1]) which agree in degree 0 by direct inspection of (1.1) and the definition of the cube complex, so it suffices to show that the right side is effaceable [5, 2.2.1]. Indeed, any sheaf F on S admits a monomorphism to a sum of sky-scraper sheaves x S j x A x, where we could even arrange that each A x is injective (this isn t necessary) so it

7 A sheaf-theoretic description of Khovanov s knot homology 7 suffices to show that the complex C (F ) has no higher cohomology when F is a skyscraper sheaf j x A. When x = this is clear beacuse then the cube complex is just A in degree 0. When x, one recognizes that the cube complex is nothing more than the complex computing the reduced simplicial cohomology (with coefficients in A) of the ( x 1)-simplex, which vanishes in all dimensions since the simplex is contractible. d There are some slight variations and generalizations of the above theorem that are useful. If S = S 1 S2 then we can form the complex C i (F ; S 1, S 2 ) := F (x) x S 2[i] with similar boundary operator. There are continuous maps: π : P(S) P(S 1 ) i : P(S 1 ) P(S) j : P(S 1 ) P(S) x x \ S 2 x x x x S 2 The first is a closed map, the second is a closed embedding, and the third is an open inclusion. The following theorem is then proved by the same method. Theorem 2.2. For the maps above we have: H i π = h i (C ( ; S 1, S 2 )), H i = H i i, H i+ S 2 j! = H i. When S = S 1 S2, we can also form the tensor product with restriction maps Ab(P(S 1 )) Ab(P(S 2 )) Ab(P(S)) (F, G ) F G (F G )(x y) := F (x) G (y) (F G ) xyx y = F xx G yy : (F G )(x y) (F G )(x y ). Clearly we have C (F G ) = C (F ) C (G ). If cohomology with supports is unappealing, reduce to computing the ordinary derived functor cohomology via the exact sequence H i (P(S), F ) H i (P(S), F ) H i (U, F U), where U is the open subset of P(S) consisting of non-empty subsets. Since global sections over the cube P(S) are just given by the stalk F, the groups in the middle vanish for i > 0 so we have H i+1 (P(S), F ) = H i (U, F U), (i > 0). d This is the combinatorial Principle of Inclusion-Exclusion.

8 8 William D. Gillam Furthermore, the exact sequence 0 H 0 (P(S), F ) H 0 (P(S), F ) H 0 (U, F U) H 1 (P(S), F ) 0 is given in terms of the complex C (F ) by 0 Ker d 0 F ( ) Ker d 1 Ker d 1 / Im d 0 0 (2.1) so we come full circle, because if the cohomology groups H i (U, F U) are computed by using the good cover U = {U s : s S}, then the Čech complex will satisfy Č i (U, F ) = C i+1 (F ) and will have the same boundary maps, with the slight homology error in degree 0 accounted for by the exact sequence (2.1). This yields another proof of Theorem Knot homology via sheaf cohomology Here we explain Khovanov s construction [11] of a knot cohomology theory in the language of sheaves on the cube TQFT Let A be the graded free abelian group with generators X (of grading 1) and 1 (of grading 1). The maps m : A A A{ 1} : A A A{ 1} ι : Z A{1} ϵ : A Z{1} 1 X X 1 1 X + X X 1 X X X X X X X 0 (the elements in tensor products of A are graded in the usual way so that the grading shifts (in {}) are necessary to make the maps grading-preserving) endow A with the structure of a graded Frobenius algebra, meaning that the following relations hold: Id A = m(ι Id Z ) (3.1) = m(id Z ι) = (ϵ Id Z ) = (Id Z ϵ) m(m Id) = m(id m) ( Id) = (Id ) m = (Id m)( Id) = (m Id)(Id ).

9 A sheaf-theoretic description of Khovanov s knot homology 9 The Frobenius algebra A is both commutative and cocommutative in the sense that both m and commute with the twist map a b b a. The Frobenius algebra A gives rise to a (graded) (1 + 1)-dimensional TQFT: a (symmetric, monoidal) functor (also called A)from the category of 1-manifolds and 2-dimensional cobordisms to Ab. The TQFT is graded in the sense that a cobordism Σ : M N yields a map A(Σ) : A(M) A(N){ χ(σ)} of graded abelian groups. The TQFT A assigns A k to a disjoint union of k circles; A assigns m and to the pair-of-pants cobordisms merging two circles to one and splitting one circle to two, respectively, and ι and ϵ for the birth (from the empty 1-manifold to the circle) and death (from the circle to the empty 1-manifold) cobordisms, respectively. For the next several sections, the particular structure of the Frobenius algebra A will not be important. However, certain additional properties of A are needed to ensure that these constructions actually give rise to knot invariants. For example, in 3.7, we will use the fact that (ι Id) : A{1} A{ 1} A A is an isomorphism. This very restrictive hypothesis is certainly not true for a general Frobenius algebra A Diagrams to sheaves Given an oriented planar diagram D of a link e with crossing set C we construct a sheaf (also called D) on the cube P(C) as follows. For x C, we resolve the crossings in x by the 1-resolution and the crossings in C\{x} by the 0-resolution (Figure 1) to obtain a collection of k disjoint circles, to which our TQFT assigns Fig. 1. The 0-resolution (left) and 1-resolution (right) of a crossing the module A k. Adjusting the grading slightly we declare D(x) := A k { x }. For x y C we can apply our TQFT to the cobordism from the x-resolution to the y-resolution given in a neighborhood of each crossing in y \ x by the obvious saddle cobordism (Figure 2) from the 0-resolution to the 1-resolution, and given by e We will see in a second that the orientation is irrelevant for a knot.

10 10 William D. Gillam Fig. 2. The saddle cobordism from the 0-resolution (top) to the 1-resolution (bottom) the identity elsewhere, to get a morphism D(x) D(y). The grading is preserved because this morphism is built from either the map m or the map at each crossing Invariance Khovanov shows [11, 5] that if the resulting graded (co)homology groups H i (P(C), D) are shifted in dimension and grading to define graded homology groups KH i (D) := H i+x(d) (P(C), D){2x(D) y(d)}, (3.2) where x(d) and y(d) are the number of crossings of x-type and y-type (respectively) in D (Figure 3) f, then the graded abelian groups KH (D) are invariants of Fig. 3. Crossings of type x (left) and y (right) the link represented by D (originally Khovanov only showed that the isomorphism classes of these groups are independent of the chosen diagram, but the stronger statement that the groups themselves are independent of the chosen diagram, up to multiplication by ±1, was later proved independently by several authors [9,12,2]. Keeping track of the shifts in grading and dimension in (3.2), this amounts to f The count of such crossings appears to depend on a choice of orientation for the link, but for knots it actually does not because one can see that reversing the orientation on both strands preserves the x- and y-type crossings. For multi-component links the effects of a relative change in orientation are minor, and can be determined in terms of the writhe number of the components.

11 A sheaf-theoretic description of Khovanov s knot homology 11 showing that the Reidemeister moves effect the unshifted homology groups H i (D) as indicated below: H i H i = H i 1 = H i { 2} {1} (R1) (R2) H i = H i 1 { 1} (R3) H i = H i (R4) As an example, we will explain how to establish invariance under (R1) in Fundamental sequence Suppose D 0, D, D 1 are three link diagrams which are the same except near one crossing c of D, where D 0 is obtained from the 0-resolution of c and D 1 is obtained from the 1-resolution. In particular, if C is the set of crossings of D, then D 0 and D 1 have crossing set C \ {c}. Let i : P(C \ {c}) P(C) and j : P(C \ {c}) P(C) be given by i(x) := x, j(x) := x {c} (as in 2.1). The map i is a closed embedding, and the map j is its open complement (so j! = j 1, c.f. 1.4). In this situation, there is a standard exact sequence 0 j! j 1 F F i i 1 F 0 (3.5) of sheaves on P(C) for any F Ab(P(C)), where both maps are the natural adjunction morphisms. (Exactness on stalks is trivial to check from the stalk formulas (1.2), (1.3).) It is immediate from the definitions of the sheaves D 0, D 1 Ab(P(C \ {c})), D Ab(P(C)) that i 1 D = D 0 and j 1 D = D 1 { 1}, so when F = D, (3.5) can be written: 0 j! D 1 { 1} D i D 0 0. (3.6) Taking cohomology with support H of (3.6) and using the equality H i j! = H i 1 of Theorem 2.2, we obtain an exact sequence 0 H 0 (D) H 0 (D 0 ) H 0 (D 1 ){ 1} H 1 (D) (3.7) called the fundamental long exact sequence. This sequence appears in the proof of [11, Prop. 9]. Here we have seen that it arises quite naturally from the sheaf-theoretic perspective.

12 12 William D. Gillam 3.5. Remark For any x P(C), the set (x \ {c}) P(C \ {c}) is the smallest among those y P(C \ {c}) such that x j(y) = y {c}. From the push-forward formula (1.2), we then see that, for any F Ab(P(C \ {c})), j F has stalks given by j F (x) = F (x \ {c}). (3.8) For x y, the specialization map (j F ) xy is nothing but F x\{c},y\{c}. From the pull-back formula (1.2), we see that j 1 (j F ) = F and i 1 (j F ) = F. The standard sequence (3.5) for j F can hence be written: 0 j! F j F i F 0. (3.9) This sequence is quite useful in light of the fact that the sheaves j F are Γ -acyclic: Lemma 3.1. H i j = 0 for all i. We give two proofs of the lemma. The first is purely sheaf-theoretic, while the second uses the cube complex in an essential way. Proof. Writing out formula (1.1) in our case, we find Γ j F = Ker (j F )( ) (j F )(y), y P(C), y where the map in question is the product of the restriction maps for j F. But one of these restriction maps is just the identity (the one where y = {c} is just Id = F : F ( ) F ( )), so we have Γ j = 0. Now, since j is the inclusion of an open set, the pushforward j is already exact, so the spectral sequence comparing the derived functors of j, Γ, and Γ j degenerates to yield H i j = R i (Γ j ). But the right hand side is zero by what we just proved. Proof. It is straightforward to check from the definitions that the cube complex C (j F ) of 2.1 is nothing but the cone on the identity map of the cube complex C (F ) of F. The Lemma now follows from the comparison Theorem 2.1 since the cone on a quasi-isomorphism is acyclic Jones polynomial If H is a finitely generated bigraded abelian group, define its Euler characteristic to be χ(h ) := ( 1) i q j rk H i j Z[q]. i,j Z Recall [10] that the Jones Polynomial of a link L can be computed in terms of the Kauffman bracket D Z[q ±1 ] of a diagram D representing it. The Kauffman

13 A sheaf-theoretic description of Khovanov s knot homology 13 bracket is uniquely defined for all diagrams by the skein-relation = q (3.10) together with its value on a simple closed loop O, which we take to be O := q+q 1, and the fact that it is multiplicative on a disjoint union of diagrams. Then the scaled Kauffman bracket K(D) := ( 1) x(d) q y(d) 2x(D) D Z[q ±1 ] (3.11) is an invariant of the link represented by D and is related to the Jones polynomial by the simple rescaling K(D) = (q + q 1 )Jones(D)(τ) τ= q. For a diagram D with crossing set C we use the shorthand χ(d) := χ(h (P(C), D)). For the usual diagram of the unknot O, we have χ(o) = χ(kh (O)) = q + q 1. Since Euler characteristic of homology groups is additive on exact sequences, it follows immediately from the fundamental sequence (3.7) that χ(h (D)) satisfies the same skein-relation satisfied by the Kauffman bracket. It also takes the same value on O and is multiplicative on a disjoint union of diagrams by the Künneth Formula because the sheaf determined by a disjoint union of diagrams is the tensor product sheaf (see the discussion after Theorem 2.2). Furthermore, the shift (3.11) used to define the scaled Kauffman bracket K(D) from D is the same as the shift (3.2) used to define χ(kh (D)) from χ(h (D)), so we have χ(kh (D)) = K(D) = (q + q 1 )Jones(D)(τ) τ= q Proof of invariance The invariance isomorphisms (R1)-(R4) of 3.3 can be naturally explained in our sheaf-theoretic language. For example, for (R1) we consider diagrams D 0, D, D 1 differing only near a crossing c of D as indicated in Figure 4. Let C be the set of Fig. 4. Diagram D 0 (left) is obtained from D (center) by the 0-resolution of c, while D 1 (right) is obtained from D by the 1-resolution of c.

14 August 4, :17 WSPC/INSTRUCTION FILE ksheaf-ws 14 William D. Gillam crossings of D, so D 0 and D 1 have crossing set C \ {c}. The diagrams give rise to sheaves D 0, D 1 Ab(P(C \ {c})) and D Ab(P(C)). Let c : D 0 D 1 { 1} ι c : D 0 { 2} D 1 { 1} be the morphisms of sheaves obtained by applying the TQFT of 3.1 to, respectively, the cobordism splitting the (fragmentary) circle in D 0 into the two circles in D 1, and the cobordism identifying the arc in D 0 with the arc in D 1 and birthing the circle in D 1. The induced map c ι c : D 0 D 0 { 2} D 1 { 1} (3.12) is easily seen to be an isomorphism by checking on stalks, where it reduces to the isomorphism mentioned at the end of 3.1. Recall the morphisms i, j : P(C \ {c}) P(C) i(x) := x j(x) := x {c} from 3.4. Consider the following commutative diagram of sheaves on P(C): 0 j! D 0 j D 0 i D 0 0 (3.13) j! c 0 j! D 1 { 1} D i D 0 0 j! ι c f = 0 j! D 0 { 2} = j! D 0 { 2} 0 0 The first row is the exact sequence (3.9) and the second row is the exact sequence (3.6). The third row is trivially exact. Only the map f requires some explanation, but it will become clear how f is defined once we describe the stalks of (3.13). To do this, we need only look at the various stalk formulas (1.2), (1.3), (3.8). If c x, then the stalk of (3.13) at x P(C) is given by: 0 D 0 (x \ {c}) = D 0 (x \ {c}) 0 0 j! c 0 D 1 (x \ {c}){ 1} f x =j! c = D(x) 0 0 j! ι c 0 D 0 (x \ {c}){ 2} = D 0 (x \ {c}){ 2} 0 0

15 August 4, :17 WSPC/INSTRUCTION FILE ksheaf-ws A sheaf-theoretic description of Khovanov s knot homology 15 If c / x, then the stalk of (3.13) at x is given by: 0 0 D 0 (x) = D 0 (x) D(x) f=id = = D 0 (x) Now it is routine tedium to check that f is well-defined, since we need only check that the above formulas for the stalks of f are compatible with specialization. Consider the map from the direct sum of the top and bottom rows of (3.13) to the middle row. On the left, the resulting map is j! of the isomorphism (3.12), and on the right the resulting map is trivially an isomorphism, so it is also an isomorphism in the middle by the Five Lemma: 0 D = j D 0 j! D 0 { 2} (3.14) Taking cohomology with support we obtain the desired equality H i (D) = H i (j D 0 ) H i (j! D 0 ){ 2} = 0 H i (j! D 0 ){ 2} = H i 1 (D 0 ){ 2} by applying Lemma 3.1 and Theorem 2.2. In general, all the knot homology isomorphisms associated to the Reidemeister moves arise from morphisms of sheaves, moving sheaves between different cubes via the functors of References [1] K. Bac lawski, Whitney numbers of geometric lattices, Adv. in Math. 16 (1975) [2] D. Bar-Natan, Khovanov s homology for tangles and cobordisms, Geometry and Topology 9 (2005) [3] R. Deheuvels, Homologie des ensembles ordonnés et des espaces topologiques, Bull. Soc. Math. France. 90 (1962) [4] J. Friedman, Cohomology in Grothendieck topologies and lower bounds in boolean complexity, arxiv.cs/ [5] A. Grothendieck, Sur quelques points d algèbra homologique, Tohoku Math. J. (2) 9 (1957) [6] A. Grothendieck et. al., Séminaire de géométrie algébrique 7: Groupes de monodromie en géométrie algébrique. (Springer-Verlag L.N.M. 288, 340, 1972/3). [7] R. Hartshorne, Algebraic Geometry. (Springer-Verlag, New York, 1977). [8] B. Iversen, Cohomology of Sheaves. (Springer-Verlag, Heidelberg, 1992). [9] M. Jacobsson, An invariant of link cobordisms from Khovanov homology, Algebr. Geom. Topol. 4 (2004)

16 16 William D. Gillam [10] L. Kauffman, State models and the Jones polynomial, Topology 26 (1987) [11] M. Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000) [12] M. Khovanov, An invariant of tangle cobordisms, Trans. Amer. Math. Soc. 358 (2006) [13] M. Khovanov, Link homology and Frobenius extensions, Fund. Math. 190 (2006) [14] K. Yanagawa, Sheaves on finite posets and modules over normal semigroup rings, J. Pure Appl. Alg. 161 (2001)