I use relationships between new invariants of links and three and fourmanifolds 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 Khovanovtheoretic 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. 

I am applying similar techniques to the recently discovered trisections of fourmanifolds and knotted surfaces. Gay and Kirby's trisection diagrams of fourmanifolds 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 want to consider the knottheoretic analogue: what can Khovanov homology tell us about Meier and Zupan's bridge trisections of knotted surfaces? In work in progress, I have constructed an Ainfinity algebra from a bridge trisection of a knotted surface. I hope to have topological applications of this gadget soon. The underlying homological theory is Szabó's extension of Khovanov homology. 

The OzsváthSzabó 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 KhovanovFloer 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 KhovanovFloer theory is functorial with respect to link cobordism. This implies that Szabó's geometric theory, the OzsváthSzabó spectral sequence, and singular instanton link homology are all functorial. I think it's also a good step towards axiomatization of these theories. I showed that the OzsváthSzabó 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 KhovanovFloer theory. 
Papers:

An annular refinement of the transverse element in Khovanov homology. Algebraic and Geometric Topology 16 (2016) 2305–2324.

Branched diagrams and the OzsváthSzabó spectral sequence . Submitted.

Strong KhovanovFloer theories and functoriality. Submitted.

Khovanov homology and trisections of knotted surfaces. In preparation.