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.

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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. Analogously, what can Khovanov homology tell us about Meier and Zupan's bridge trisections of knotted surfaces? I have constructed a combinatorial invariant of knotted surfaces using an extension of Khovanov homology due to Szabó. The invariant distinguishes the unknotted sphere from a certain family of knotted spheres. This invariant is extracted from a certain A-infinity algebra associated to a bridge trisection of a surface. This algebra may be helpful in recognizing when bridge trisections can be destabilized. A draft of this work is available here.

This project can also be seen as a "toy model" for connections between Heegaard Floer theory and trisections of four-manifolds. In future work I plan to develop the analogue of the link homology invariant to four-manifolds.

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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 link homology theory, the Ozsváth-Szabó spectral sequence, and singular instanton link homology are all functorial. Functoriality of Szabó's geometric theory is a crucial element of the knotted surface invariants above.

I used these techniques to show that all Khovanov-Floer theories over a certain ring are mutation-invariant.

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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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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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