Adam Saltz

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I use relationships between new invariants of links and three- and four-manifolds to better understand knots, knotted surfaces, and contact structures. My research is motivated by connections between Khovanov homology and Heegaard Floer homology first observed by Ozsváth and Szabó. With Diana Hubbard, I constructed an extension of Plamenevskaya's invariant of transverse links in Khovanov homology. Our definition is analogous to Hutchings, Latschev, and Wendl's definition of algebraic torsion in embedded contact homology and Heegaard Floer homology. We show that our invariant solves the word problem in braid groups, and we use it to disprove a conjecture about the length of a certain Khovanov-theoretic spectral sequence. I have written a computer program to calculate the invariant, available here. Numerical evidence suggests that a related invariant is an effective invariant of transverse links.

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I am applying similar techniques to the recently discovered trisections of four-manifolds and knotted surfaces. Gay and Kirby's trisection diagrams of four-manifolds are closely related to Heegaard diagrams, so it's natural to try to use techniques from Heegaard Floer homology to study them. Before tackling the considerable technical complications arising from such a project, I consider the knot-theoretic analogue: what can Khovanov homology tell us about Meier and Zupan's bridge trisections of knotted surfaces? In recent work I construct an invariant of knotted surfaces with values in Z/2Z which can be computed from a triplane diagram. This invariant distinguishes the unknotted sphere from an infinite family of knotted spheres. I also construct an A-infinity algebra which is not quite an invariant of the surface but which may be useful in understanding stabilization of bridge trisections. These invariants should have analogues in trisections of four-manifolds via Heegaard Floer homology.

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The Ozsváth-Szabó spectral sequence is one of many (at least eight!) spectral sequences from Khovanov homology to an invariant of links or manifolds. Following work by Baldwin, Hedden, and Lobb, I defined strong Khovanov-Floer theories, i.e. gadgets which assign, to a link, a filtered complex whose associated spectral sequence extends from Khovanov homology. I showed that every strong Khovanov-Floer theory is functorial with respect to link cobordism. This implies that Szabó's geometric theory, the Ozsváth-Szabó spectral sequence, and singular instanton link homology are all functorial. I applied these techniques to show that each of those theories is invariant under link mutation. In ongoing work with John Baldwin and Cotton Seed I use similar techniques to construct a tangle theory for Szabó's geometric spectral sequence. I think it's also a good first step towards axiomatization of these theories and applications to contact topology.

I showed that the Ozsváth-Szabó spectral sequence can be constructed from simple Heegaard diagrams with a more obvious connection to Khovanov homology. It is easy to "see" the transverse and contact invariants in these diagrams. They also play a role in showing that Heegaard Floer homology produces a strong Khovanov-Floer theory.

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