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Research Statement: Victor Summers

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Research Statement: Victor Summers Research Statement: Victor Summers My research lies at the confluence of knot 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 Jones Polynomial [8] which sparked a new era in the theory of knots 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 homology 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) link 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 ∼ p1(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 torus knot 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 Floer homology (a categorification of the Alexander polynomial). 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 Euler characteristic of its Khovanov homology Kh∗;∗(L): i j i; j JbL(q) = ∑ (−1) q rank(Kh (L)) i; j2Z 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 unknots [10] : A knot K is an unknot 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 unlinks [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(G) cobordism G : L1 ! L2 (i.e. an oriented manifold W with ¶W = −L1 t L2) Kh(L1) Kh(L2) induces a homomorphism Kh(G) : Kh(L1) ! Kh(L2) of Khovanov homology groups. There are no maps between Jones polynomials. cq cq 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 slice genus 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 Hopf link, 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 gauge theory, 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 W0;W1;:::;W6 – whose details I will omit here. Theorem 1. The Khovanov homology for braids in W0;W1;W2;W3; 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 W4 - W6, 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 W0 - W3 rely heavily on local Z2 homological thinness (for each homological grading, Khovanov homology is supported on at most two adjacent polynomial gradings).
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