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See my research statement for a description of some of my past and upcoming projects.
On naturality of the Ozsvath-Szabo contact invariant, with Hedden
We discuss functoriality properties of the Ozsvath-Szabo contact invariant, and expose a number of results which seemed destined for folklore. We clarify the (in)dependence of the invariant on the basepoint, prove that it is functorial with respect to contactomorphisms, and show that it is strongly functorial under Stein cobordisms.
published in Transactions of the AMS
We show that a quasipositive surface with disconnected boundary induces a map between the knot Floer homology groups of its boundary components preserving the transverse invariant. As an application, we show that this invariant can be used to obstruct decomposable Lagrangian cobordisms of arbitrary genus within Weinstein cobordisms. The construction of our maps rely on the comultiplicativity of the transverse invariant. Along the way, we also recover various naturality statements for the invariant under contact +1 surgery.
We define combinatorial invariants of Legendrian and transverse links in universally tight lens spaces using grid diagrams, generalizing [OST08] and prove that they are equivalent to the invariants defined in by Baldwin, Vela-Vick and Vertesi and also Lisca, Ozsvath, Stipsicz and Szabo. Using these invariants, we give a new contact geometric characterization of simple knots in lens spaces and discuss a potential application to the Berge conjecture.
published in Journal of Symplectic Geometry
We show that the transverse invariant of a pointed monodromy having fractional Dehn twist coefficient greater or equal to one is non-zero. This generalizes work of Honda, Kazez and Matic and also work of Plamenevskaya.
We construct F[U]-module homomorphisms of HFK minus groups associated to concordances between knots in the 3-sphere.
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