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Home , Khovanov homology

Research Statement: Victor Summers

My research lies at the confluence of theory, graph theory, low-dimensional topology and homological algebra, with an overarching theme of categorification. Categorification is a powerful method for structurally enriching algebraic invariants, brought into sharp focus with M. Khovanov’s publication of his 1999 paper A Categorification of the [8] which sparked a new era in the theory of and links. In addition to increasing the potency of invariants, this new paradigm has shed light on previously unseen connections between links, graphs, manifolds and other related areas such as combinatorics, geometry and mathematical physics. I have focused on the Khovanov of links and the magnitude homology of graphs, which may be viewed as categorifications of their Euler characteristics; the Jones polynomial and magnitude power series, respectively. Torsion, that is, elements of finite order, is a phenomenon appearing at the homological level making no appearance at the polynomial level. The structure and topological significance of torsion are still largely a mystery. My aim has been to aid in its demystification—determining which other T(p,q) invariants it relates to and how. One of my main results is an extension of A. Shumakovitch’s result on the existence of torsion of order 2 in Z2- homologically thin links, to a class of links that are "locally" homologically thin; this class includes the torus links T(3,q); so called because they ∼ π1(L(p,q)) = p can be laid on the surface of a torus without self-intersection. This work Z rests on a classification of 3-braids due to K. Murasugi, and uses the long exact sequence in Khovanov homology along with a careful analysis of interactions between the spectral sequences of Bockstein, Turner, Lee and G(p,q) Bar-Natan. In magnitude homology, I have formulated a bound on possible torsion sizes that depends on geodesic counts when viewing graphs as finite metric spaces. Kaneta and Yoshinaga showed that the homology of simplicial complexes embeds into the magnitude homology of certain associated graphs, and I have used this idea to prove the aforementioned bound to be sharp, and also to construct families of graphs with a prescribed size of torsion with the aid of Pachner moves on triangulations of lens spaces. A previously unexplored connection between knots, graphs and manifolds that made itself known in the course of my research is the following: The T(p,q) gives rise to the lens space L(p,q), whose fundamental group is Zp, and whose triangulations give rise to graphs with Zp-torsion in magnitude homology. I hope to flesh out the details of this connection in the future.

1 Invariants and their Structure

A common tool used in the study of knots, graphs, and manifolds is that of an invariant—typically the assignment of a number, polynomial or group to objects in such a way as to encode information about those objects. The more structured an invariant, the more information it may carry. To illustrate this principle, consider the shapes on either side of this paragraph. While undistinguished by the cardinality of their symmetry groups (both being of size four), they are nonetheless distinguished by the algebraic structure of these groups: Klein-4 group on the left, cyclic group of order four on the right. In 1999, Khovanov sparked a new era in the algebraic enrichment of invariants: categorification. Most generally, categorification is the promotion of a n-category to (n+1)-category. In practice, this often involves the upgrading of a polynomial invariant, which are ubiquitous, to a collection of groups from which the polynomial can be recovered. Intriguingly, connections appear at the categorified level where no such connections exist at the polynomial level. These connections often appear in the form of a spectral sequence; a sequence of groups whose first term is one homology theory which converges to another. For example, there is no way to compare the Jones and Alexander polynomials, but there is a spectral sequence from Khovanov homology to instanton knot (a categorification of the ).

Khovanov Homology

In 1983, V. Jones discovered an invariant of links now known as the Jones polynomial. In 1990, Jones was subsequently awarded the Fields Medal for his work. Though a powerful invariant in its own right, this polynomial was substantially upgraded to a (finitely generated) bigraded abelian group by Khovanov in his seminal paper, A Categorification of the Jones Polynomial [8]. The Jones polynomial JbL(q) of a link L may be recovered as the graded of its Khovanov homology Kh∗,∗(L):

i j i, j JbL(q) = ∑ (−1) q rank(Kh (L)) i, j∈Z Khovanov homology has many advantages over the Jones polynomial:

1. Khovanov homology distinguishes more links than the Jones polynomial. For example, the knots 51 and 10132 are not distinguished by the Jones polynomial but are distinguished by Khovanov homology; ∼ J51 (q) = J10132 (q), while Kh(51) =6 Kh(10132). 2. Khovanov homology detects [10] : A knot K is an if and only if it has the Khovanov homology of an unknot. It is currently unknown whether the Jones polynomial detects unknots. 3. Khovanov homology detects [4], while the Jones polynomial does not. In fact, there are many non-trivial k-component links with the Jones polynomial of the trivial k-component link [2].

2 4. Khovanov homology is functorial with respect to cobordisms. That is, a Kh(Γ) cobordism Γ : L1 → L2 (i.e. an oriented manifold W with ∂W = −L1 t L2) Kh(L1) Kh(L2) induces a homomorphism Kh(Γ) : Kh(L1) → Kh(L2) of Khovanov homology groups. There are no maps between Jones polynomials. χq χq 5. Through Rasmussen’s s-invariant—derived from a spectral sequence whose first page is the Khovanov homology—functoriality gives rise to a lower ? bound on the of knots. This bound cannot be extracted form the JL1 (q) JL2 (q) Jones polynomial, and provides for a substantially shorter proof of Milnor’s conjecture on the slice genus of (p,q)-torus knots [15], which was first proved using a sophisticated gauge theoretical argument [9].

Torsion in Khovanov is currently not well understood. There are at least two compelling reasons to study patterns of torsion in Khovanov homology. First, structural results on torsion have the potential to yield results concerning the strength of Khovanov homology. The Shumakovitch conjecture, if true, would be a prime example. Shumakovitch conjectures that the Khovanov homology of every link contains Z2-torsion, with the exceptions of the unknot and , along with their connected sums and disjoint unions [16]. Since the only link on this list of exceptions with the Khovanov homology of an unknot is the unknot itself, the Shumakovitch conjecture would imply that Khovanov homology is an unknot detector; again, the original proof of this fact involved , but this would provide a simpler, purely combinatorial proof. The Shumakovitch conjecture is now known to be true for various classes of links ([1],[14]). Second, torsion may encode significant topological information about links. While it is possible that torsion is little more than an artifact of the construction, my recent results suggest otherwise. Alexander’s theorem states that every link is the closure of an n-stranded braid for some n. To the left is depicted a 3-stranded braid along with its closure, a link. The braid Przytycky-Sazdanovic (PS) conjecture posits a relationship between the sizes of closure torsion in Khovanov homology and the braid index of a link; the least value of n for which a link is the closure of an n-stranded braid. In Theorem 1, we give a partial positive answer to the first part of the PS conjecture, which states that the Khovanov homology of a class of closed 3-braids may contain only Z2-torsion. This result rests on a classification of 3-braids due to K. Murasugi [13], which says that each closed 3-braid belongs to exactly one of seven sets Ω0,Ω1,...,Ω6 – whose details I will omit here.

Theorem 1. The Khovanov homology for braids in Ω0,Ω1,Ω2,Ω3, contains only Z2-torsion.

This hard-won result was obtained through the use of the powerful homological algebraic tool of spectral sequences. Fundamentally, the proof boils down to two steps: 1) Show that these links have no odd order torsion in Khovanov homology, and 2) show that all even order torsion is in fact of order 2 by arguing that the Z2-Bockstein spectral sequence collapses on the second page. Step 1 was shown through induction arguments using the skein long exact sequence in Khovanov homology, while step 2 was shown in a more indirect manner by taking advantage of relations between the Bockstein spectral sequence and various other spectral sequences arising from alternative differentials on the Khovanov complex. In particular, this theorem says that all torsion in three-stranded torus links is of order two. Computations heavily suggest that this result extends to all three-stranded braids.

3 This research is part of an ongoing project aimed at characterizing torsion in the Khovanov homology of braid representations. The methods of proof outlined here do not immediately follow through for the 3-braids of types Ω4 - Ω6, but it is hoped that they may be adjusted or extended in order to state a general a result as possible.

Future Directions

1. The spectral sequence arguments used to show that there is only torsion of order 2 in the Khovanov homology of 3-braids of types Ω0 - Ω3 rely heavily on local Z2 homological thinness (for each homological grading, Khovanov homology is supported on at most two adjacent polynomial gradings). Types Ω4 - Ω6 do not, in general, have this property.

(a) Can we adjust this method to show that braids of types Ω4 - Ω6 only have torsion of order 2? (b) Can we make local statements about torsion in these remaining 3-braid types using Z2 homological thinness in certain parts of their Khovanov homology? 2. Algebraic Morse Theory provides a method for simplifying homology calculations. More specifically, for a given chain complex (C,∂) we obtain a weighted, directed graph G with a directed edge (x,y) of weight α whenever ∂x = αy. Given a matching M on G, the graph GM is obtained by reversing all arrows in M. If GM is acyclic, then the (Morse) complex (CM,d) obtained by omitting all vertices in the edges of M, and with an appropriate differential d, is homotopy equivalent to (C,∂). With potentially many fewer generators, the homology of this new complex is often substantially easier to compute. This method has been successfully used by Y. Gu [3] to calculate the magnitude homology groups of cycle graphs. (a) Can we find classes of link diagrams whose associated weighted-directed graphs admit matchings whose associated Morse complexes have relatively easy-to-compute homology groups? (b) Are there link diagrams whose associated weighted-directed graphs admit matchings whose associated Morse complexes can be used produce further statements regarding the structure of torsion?

Magnitude Homology

Many mathematical constructions come equipped with a canonical measure of size: the cardinality of a set; the Euler characteristic of a topological space; the dimension of a . T. Leinster added the magnitude of an enriched category to this list [11]. Since a metric space may be viewed as an enriched category, and a finite graph may be viewed as a metric space when equipped with the shortest path metric, one is naturally lead to a notion of magnitude for finite graphs. By considering all positive scalings tM of a metric space M, we obtain the magnitude function. In a sense, the magnitude may be thought of as the effective number of points because magnitude approaches cardinality as t tends toward infinity. In general, this function may be fairly erratic, possibly having singularities and negative values. In the case of graphs it takes the convenient form of a n power series with integer coefficients: #G(q) = ∑n≥0 anq ∈ Z[[q]].

{Graphs} ⊂ {Metric Spaces} ⊂ {Enriched Categories}

4 The magnitude invariant is capturing genuinely different information to the most well-known graph invariants. For instance, there exist graphs with the same cycle matroid and thus the same Tutte polynomial, same number of proper vertex colorings, same minimal cycle length, and so on, but are distinguished by their magnitude. On the other hand, there are polynomials with the same magnitude that are nonetheless easily distinguished by other, simple invariants [12]. Another important question to ask of graph invariants is whether they see Whitney twists. Willerton has shown that magnitude is not invariant under Whitney twists—hence cannot be a specialization of the Tutte polynomial—but is invariant under Whitney twists along adjacent vertices. A clue that a polynomial invariant may be realized as the Euler characteristic of some homology theory is an alternating sum formula. For example, inspired by an alternating state-sum formula, L. Helme-Guizon and Y. Rong upgraded the chromatic graph polynomial χq(G) to chromatic graph homology HGR(G), whose graded Euler characteristic is χq(G) [5]. Similarly, Hepworth and Willerton used an alternating formula of Leinster in [12] as impetus for their construction of magnitude graph homology [6]. Magnitude homology MH∗,∗(G) is a bigraded abelian group whose graded Euler characteristic is the magnitude power series:

k ` #G(q) = ∑ (−1) q dimQ(MHk,`(G) ⊗ Q) k,`≥0 Various properties of the magnitude power series are "shadows" of corresponding properties of the homology theory: Multiplicativity of magnitude with respect to the Cartesian product of graphs is upgraded to a Künneth short exact sequence; additivity with respect to disjoint union is lifted to a direct sum formula in homology; an inclusion-exclusion formula (for projecting decompositions) is upgraded to a Mayer-Vietoris-type sequence.

0 MH∗,∗(G ∩ H) MH∗,∗(G) ⊕ MH∗,∗(H) MH∗,∗(G ∪ H) 0

χq

#(G ∪ H) = #G + #H − #(G ∪ H)

Figure 1: An inclusion-exclusion formula lifts to a Mayer-Vietoris-type sequence in homology.

Upon constructing magnitude homology, Hepworth and Willerton made various observations and conjectures based on computations performed with the aid of a computer program. They show that if graphs G and H are MH-diagonal, i.e. if their groups vanish away from the diagonal k = `, then their join G?H is also MH-diagonal; they give an explicit formula for the homology of trees, they conjecture an explicit formula the homology groups of the cycle graphs Cn (since confirmed by Y. Gu [3]), and much more. It is currently unknown as to whether magnitude homology is invariant under vertex-adjacent Whitney twists. In these computer calculations, Hepworth and Willerton found no torsion subgroups. In attempting to prove that magnitude is torsion free, I discovered the following bound; this bound, in particular, implies that the closer to the main diagonal you get, the smaller the possible sizes of torsion. Theorem 2. Let G be a graph. For vertices x,y of G, let g(x,y) denote the number of geodesic paths in G from x to y. Then, MHk,`(G) contains no elements of order p for k−1 ! p > max ∑ (d(xi,xi+1) − 1)g(xi,xi+1) . (x0,...,xk)∈MCk,`(G) i=0

5 In 2017, Kaneta and Yoshinaga constructed a graph with elements of order 2 in magnitude homology [7]. They showed that the singular homology groups of a simplicial complex embed into the magnitude homology groups of an associated graph. I have since used this embedding to show that the bound of Theorem 2 is sharp. Additionally, Pachner moves on triangulations of lens spaces have lead me to the following result. Theorem 3. For a given prime p, there exist infinitely many isomorphism classes of graphs whose magnitude homology contains torsion of order p.

P1 P2

−1 −1 P1 P2

Figure 2: Pachner moves on simplicial complexes of dimension 2. This theorem is in stark contrast to current knowledge on torsion in Khovanov homology, where there is no known method for constructing links whose Khovanov homology contains elements of a prescribed order. Future Directions

1. There are many numerical graph invariants for which we only have bounds (e.g. minimal crossing numbers), and invariants that we can only detect with the aid of a computer algorithm (e.g. outer-planarity—a graph is outer-planar if it is planar and each vertex lies on an outer edge). Given that magnitude homology detects genuinely different information to the most well-known graph invariants, we should investigate its bearing on these bounds and algorithms. For example, does magnitude homology detect outer-planarity?

2. The main diagonal in magnitude homology (the groups MH`,`(G)) forms an invariant in its own right, albeit weaker than magnitude homology itself. (a) What properties of a graph are encoded in the main diagonal? Computations suggest the main diagonal is, in part, counting cyclic subgraphs. (b) Can we characterize the main diagonal, at least for certain families of graphs? 3. As illustrated in Figure 2 for simplicial complexes of dimension 2, Pachner moves send a tringulation of a manifold to another triangulation of the same manifold. As Kaneta and Yoshinaga point out, the simplicial homology of such triangulations embed into the magnitude homolgogy of their associated graphs. (a) This fact can be used to produce many graphs with a prescribed size of torsion. Is it possible to characterize the effect of Pachner moves on magnitude homology? (b) Can we characterize the effect of Pachner moves at the level of the associated graphs? (c) Is there a graph with torsion in magnitude homology which does not stem from the embedding of Kaneta and Yoshinaga? If no, then we have a classification of graphs with torsion in magnitude homology, namely thsoe arising from triangulated manifolds. If yes, can we characterize those graphs with torsion in magnitude homology? 4. Can we use the aforementioned methods of algebraic Morse theory to make explicit calculations of families of graphs apart from the cyclic graphs?

6 References

[1] M. M. Asaeda and J. H. Przytycki. Khovanov homology: torsion and thickness. In Advances in topological quantum field theory, pages 135–166. Springer, 2004. [2] S. Eliahou, L. H. Kauffman, and M. B. Thistlethwaite. Infinite families of links with trivial jones polynomial. Topology, 42(1):155–169, 2003. [3] Y. Gu. Graph magnitude homology via algebraic morse theory. arXiv preprint arXiv:1809.07240, 2018. [4] M. Hedden and Y. Ni. Khovanov module and the detection of unlinks. Geometry & Topology, 17(5):3027– 3076, 2013. [5] L. Helme-Guizon and Y. Rong. A categorification for the chromatic polynomial. Algebraic & Geometric Topology, 5(4):1365–1388, 2005. [6] R. Hepworth and S. Willerton. Categorifying the magnitude of a graph. arXiv preprint arXiv:1505.04125, 2015. [7] R. Kaneta and M. Yoshinaga. Magnitude homology of metric spaces and order complexes. arXiv preprint arXiv:1803.04247, 2018. [8] M. Khovanov. A categorification of the jones polynomial. arXiv preprint math/9908171, 1999. [9] P. B. Kronheimer and T. S. Mrowka. Gauge theory for embedded surfaces, i. Topology, 32(4):773–826, 1993. [10] P. B. Kronheimer and T. S. Mrowka. Khovanov homology is an unknot-detector. Publications mathéma- tiques de l’IHÉS, 113(1):97–208, 2011. [11] T. Leinster. The magnitude of metric spaces. Documenta Mathematica, 18:857–905, 2013. [12] T. Leinster. The magnitude of a graph. In Mathematical Proceedings of the Cambridge Philosophical Society, pages 1–18. Cambridge University Press, 2017. [13] K. Murasugi. and its applications. Springer Science & Business Media, 2007. [14] J. H. Przytycki and R. Sazdanovic. Torsion in khovanov homology of semi-adequate links. arXiv preprint arXiv:1210.5254, 2012. [15] J. A. Rasmussen. Khovanov homology and the slice genus. arXiv preprint math/0402131, 2004. [16] A. Shumakovitch. Torsion of the khovanov homology. arXiv preprint math/0405474, 2004.

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