Khovanov homology and torsion

: This article offers an introduction to Khovanov homology with a brief overview of the developments in low dimensional topology it has inspired, and relations with other link homology theories. The emphasis is placed on the role of torsion in Khovanov homology and several open problems.

Among the first of many notable applications of Khovanov homology we emphasize J. Rasmussen's [37] concordance invariant, called the s-invariant, that enabled him to give a purely combinatorial proof of the Milnor conjecture, also known as the Kronheimer-Mrowka theorem. Theorem 6.1.1 (J. Rasmussen). The s-invariant provides a lower bound on the slice genus of a knot. Theorem 6.1.2 (Milnor conjecture). The slice genus of the T(p, q) torus knot is equal to 1 2 (p − 1)(q − 1).
Using the same s-invariant A. Shumakovitch improved the slice-Bennequin inequality for some classes of knots [40]. Khovanov's construction led to many significant advances and new discoveries. There are several extensions of Khovanov homology, including odd and reduced Khovanov homology, as well as symplectic and bordered Khovanov homology which have been studied since. Other knot polynomials such as HOMFLYPT, Kauffman 2-variable, and sl(N) link polynomial have also been categorified. Motivated by the Floer homology, Ozsváth, Szabó, and Rasmussen constructed knot Floer homology that categorifies the Alexander polynomial. Soon after, this categorification was given a purely combinatorial description by Manolescu, Ozsvath and Sarkar [30]. This abundance of new structures has provided many answers to classical questions in 3-and 4dimensional topology.

Definition and structure of Khovanov link homology
Khovanov link homology is a functor that assigns to every link L bigraded homology groups Kh(L) whose Euler characteristic is the Jones polynomial of L, and to link cobordisms homomorphisms of homology groups. Here we define Khovanov homology in broad strokes, mostly relying on Bar Natan's cube construction [3]. For details check one of the following excellent resources [1, 3,44,45,43], each of which emphasizes different aspects of Khovanov link homology. This construction is based on the Kauffman state sum expression for the Jones polynomial, known as the Kauffman bracket polynomial [19]. This formula can be used instead of the skein relation to express the Jones polynomial as a sum of powers of binomials (q + 1 q ), which is the Jones polynomial of the unknot. Let D be an n-crossing diagram of a knot K. A Kauffman state s of the knot diagram D is a function from the set of crossings {C i } n i=1 of D to the set {0, 1}. Diagrammatically, we assign to each crossing of D a marker according to the convention in 6.1, where each crossing is replaced by two parallel arcs. The Kauffman state can be viewed as a collection of circles Ds embedded in the plane obtained by smoothing each crossing in D. Let |Ds| be the number of circles in Ds.
There are 2 n Kauffman states that can be conveniently organized as the vertices of an n-dimensional unit cube I n = [0, 1] n . Each vertex V ∈ I n of the cube corresponds to one Kauffman state s: the state that associates a 0 or 1 smoothing to a crossing C i depending on whether the i-th coordinate of that vertex is 0 or 1.

A ⊗2
A ⊗2 → → Let A 2 = Z[x]/(x 2 = 0) be a graded module with 1 in bidegree (0, 1) and x in bidegree (0, −1). A 2 has the Frobenius algebra structure including the multiplication and comultiplication maps: To each vertex assign a module C(Ds) = C(D(V)) = A  [3]. Next, let e be the edge of the cube I n between vertices V and W whose coordinates are the same except in one place where W has entry 1 and V has 0. We define the per edge map de : C(D(V)) → C(D(W)) for each such edge e in the following way: -if the number of Kauffman circles in the Kauffman state of vertex W is smaller than the number of circles in the state associated to V then de is a multiplication on the corresponding copies of A 2 -otherwise de is a comultiplication ∆ on the respective tensor factors and all other tensor factors are mapped by the identity map.
The structure we have described is referred to as the cube of resolutions and the Khovanov chain complex is its total complex, Figure 6 Khovanov homology also appears in mathematical physics in the work of Gukov, Schwarz, Vafa, and Witten, where it admits a description in terms of so-called BPS states.

Torsion of Khovanov link homology
Recall that the Jones polynomial equals the Euler characteristic of Khovanov homology, so the Jones polynomial depends only on the free part of Khovanov homology. This section is devoted to results concerning torsion of Khovanov homology intertwined with the ideas and methods that could be useful in advancing our understanding of torsion and the additional information it may carry about knots and their cobordisms. Although computations hint at the abundance of torsion, describing and understanding torsion in Khovanov homology is still elusive. A great deal has been written about torsion of order 2; however, torsion of higher order is still a mystery [3,5,11]. Torsion in Khovanov homology is understood only for some classes of knots and links [2, 34,36,41,42]. As another illustration of the importance of torsion in Khovanov homology, note that the affirmative resolution of Shumakovitch's conjecture would provide a new proof that the Khovanov homology is an unknot-detector. The only currently known proof of this fact due to P. Kronhaimer and T. Mrowka [27] relies on the relations with the instanton Floer homology.

Approach 1. Spectral sequences for constructing odd torsion in link homology
A first step to understanding the torsion in Khovanov homology is to construct links whose Khovanov homology has torsion of a prescribed order. A. Shumakovitch recently used the Bockstein spectral sequence to compute all of the torsion in the Khovanov homology of alternating knots. In particular, he proves Conjecture 6.3.1 for alternating knots and shows that Khovanov homology of thin knots has only Z 2 -torsion. D. Bar-Natan showed that Khovanov homology of the T(8, 7) torus knot contains Z 7 , Z 5 , Z 4 , and Z 2 -torsion. This 48 crossing knot reaches the limits of current computational resources [26,6]. Extending theoretical results to non-alternating links in order to find Zp-torsion for odd primes p will require either a more sophisticated spectral sequence or perhaps arguments of a different nature altogether.

Approach 2. Relations between Khovanov link, Hochschild algebra and chromatic graph homology theories
Hochschild homology is a well-understood cyclic homology theory of associative algebras. Przytycki observed, somewhat surprisingly, that Khovanov homology and Hochschild homology share a common structure [35]. The limit of the Khovanov homology of T(2, n) torus links when n is large can be interpreted as the Hochschild homology of the algebra A 2 = Z[x]/(x 2 = 0) and some of the torsion of Hochschild homology can be seen in Khovanov homology of T(2, n).
The cyclic nature of Hochschild homology can be interpreted pictorially. The n-th chain group can be pictured as an n-cycle. The differential corresponds to contracting the edges of n-cycle one by one to get an (n − 1)-cycle and multiplying the elements in the copies of A labelling the endpoints of the contracted edge. The graph Γ(D A ) corresponding to the T(2, n) torus link and Kauffman state sending all crossings to 1 is just an n-cycle. The chromatic graph homology HGR A2 (Γ) is one of several categorifications of polynomial graph invariants, and one of two categorifications of the chromatic polynomial [10,15]. Given a graph Γ the chromatic graph polynomial P(Γ) satisfies the recursive deletion-contraction relation where Γ − e is the graph obtained by deleting an edge e from a graph Γ, and Γ/e is the graph obtained by contracting the same edge. After categorification, this relation lifts to long exact sequence of chromatic graph cohomology groups [15]: The construction of HGR A2 (Γ) follows that of Khovanov: a bigraded homology theory HGR i,j A2 (Γ) is associated to a graph Γ and a commutative (graded) algebra A 2 , in such a way that its (graded) Euler characteristic is the value of the chromatic polynomial at the graded dimension of the algebra A 2 .
In terms of the chromatic graph homology, the relation between Khovanov and Hochschild homology can be expressed by saying that the chromatic graph homology of the cycle of length n is the Hochschild homology of algebra A 2 = Z[x]/(x 2 = 0) through a range of dimensions increasing with n. Theorem 6.3.2 (Correspondence between Khovanov and chromatic homology). Let Γ(D A ) be the graph associated to diagram D of a link L whose girth, the length of the shortest cycle, ℓ = ℓ(G) > 1. Then the first ℓ Khovanov homology Kh(L) groups of L and chromatic graph homology HGR A2 (Γ(D A )) have isomorphic torsion [34].
This point of view enables us to describe some Z 2 -torsion in the Khovanov homology of certain classes of links via torsion in chromatic homology [2,14,34,36] expressed in terms of combinatorial data of one of the corresponding graphs. For example, the torsion depends on the cyclomatic number that is the rank of the first homology of that graph as a 1-dimensional CW-complex. torH n−4,n+2|Ds This proposition establishes Shumakovitch's conjecture for all adequate and semi-adequate knots whose corresponding graphs have odd cycles. Chromatic homology, like Khovanov homology of alternating links, is supported only along two diagonals [9], with torsion appearing only on one. An interesting computational problem would be to explicitly compute all torsion along the whole diagonal in chromatic homology; pulling this information back via Theorem 6.3.2 would greatly extend our current knowledge of the torsion in Khovanov homology.

Approach 3. Compute torsion via Khovanov tangle invariants
In what we have described so far, computing Khovanov homology requires global knowledge of the topology of Kauffman states. Instead, one can take a different approach and decompose a link diagram into local pieces called tangles. While it is straightforward to piece together tangles to obtain a link, it is quite challenging to piece together any of the existing Khovanov tangle invariants [21,22,4] to obtain the Khovanov homology of the link. Roberts [38,39] constructs a Khovanov tangle invariant whose algebraic structure is inspired by the bordered Heegaard Floer homology. Most importantly, Roberts obtains gluing formulas which he uses to construct the Khovanov homology of a link from the Khovanov invariants of the tangles in its decomposition. This approach may be most suitable for studying the Khovanov homology of classes of links with especially nice tangle decompositions, such as almost-alternating links and closed 3-braids.

Goal 2. Describe topology of the link in terms of torsion of Khovanov homology
With a discovery of the new knot invariant one of the most intriguing questions is determining how much it can tell us about knots. In our framework, this translates into the question: what does torsion in Khovanov homology reveal about topological properties of the link. As mentioned above, it is conjectured that the existence of torsion in the Khovanov homology of a knot implies the knot is nontrivial. Moreover, computations indicate a potential relationship between the existence of Zp-torsion in the Khovanov homology of a link and the braid index of the link. However, the infinite family of links with braid index four containing Z 2 s -torsion with s ≤ 23 constructed in [31] provides a counterexample to the PS-conjecture [36] and hints that such a relation, if it exists, will be more subtle.

Homological invariants of alternating and quasi-alternating cobordisms
Khovanov homology is functorial, up to overall minus sign, under link cobordisms. However, in general, it is extremely difficult to compute homomorphisms of Khovanov homology induced by link cobordisms.
Notice that the number of spanning trees s(Γ) in a graph Γ satisfies a recursive relation similar to the deletion-contraction formula for the chromatic polynomial Spanning trees of Γ(D A ) play a fundamental role in link homology [8,46]. The reduced Khovanov homology of L can be computed via a complex generated by spanning trees of Γ(D A ) but the combinatorial form of the differential is currently unknown. The same property holds for the knot Floer homology of L. When D is a minimal alternating diagram, differentials in both complexes are zero, and the rank of reduced Khovanov and knot Floer homologies of L equals the number of spanning trees in Γ(D A ), and also the determinant of L [32].

Goal 3. Compute homomorphisms of Khovanov homology induced by "alternating" cobordisms between alternating links
The spanning tree model gives a good inductive realization of Khovanov homology, and maps induced by alternating cobordisms could be effectively computable. The alternating cobordisms are those generated by the cobordisms illustrated in 6.4: they arise from smoothing a crossing, saddles, and births and deaths of circles.
Khovanov-Rozansky sl(n) link homology [24,25] and its variation due to Mackaay, Stošić, Vaz [28], are based on homology theories of planar graphs, functorial under foam cobordisms (in R 2 × [0, 1]) between these graphs. Similarly one could search for an interpretation of alternating cobordisms, which are realized by surfaces embedded in R 4 , in terms of cobordisms between checkerboard graphs of alternating diagrams. Cobordisms between planar graphs induced by alternating cobordisms will resemble foams. The same idea could be employed towards computing maps between link Floer homologies of alternating links induced by alternating cobordisms, or the maps induced by lifts of alternating cobordisms to the double branch cover.

Goal 4. Find a combinatorial framework for quasi-alternating knots
Quasi-alternating links are generalizations of alternating links, defined in an inductive way.
The set Q of quasi-alternating links is the smallest set of links including the unknot and satisfying the following property: If a link L has a diagram D with a crossing c such that both smoothings, L 0 and L 1 , of c are in Q, and det(L)=det(L 0 )+det(L 1 ), then L is in Q [33,7,29].
The notion of quasi-alternating link is hard to capture combinatorially. The initial search for quasi-alternating link families was performed using the knot theory software LinKnot [17,16] and the computational results have already been used to show that certain knots and links are not quasi-alternating [12]. It would be very exciting to find a combinatorial model for some large class of quasi-alternating links, incorporating inductive steps in their construction. Such a combinatorial model should also give a generalization of planar graphs (checkerboard graphs of alternating link diagrams) to more subtle higher-dimensional structures.
Furthermore, the Khovanov and knot Floer homology of quasi-alternating links are easy to compute, due to the vanishing of the differentials in short exact sequences that build up to homology groups of these links. Quasi-alternating cobordisms between quasi-alternating links can be defined as compositions of cobordisms built out of quasi-alternating crossings.

Acknowledgment:
The author would like to thank S. Jablan, M. Khovanov, and J. Przytycki for their support and guidance that have been invaluable, and A. Lowrance for helpful conversations and sharing his ideas about Goal 1, Approach 6.3. Many thanks to the anonymous referee for corrections.