Topology

Satellite operations are a valuable method of constructing complicated knots from simpler ones, and much work has gone into understanding how knot invariants change under these operations. We describe a new way of computing the (UV=0 quotient of the) knot Floer complex using an immersed Heegaard…

We state our recent result determining the quantum filtration structure of the Lee homology of torus links. This implies a relative adjunction-type inequality for s-invariants of two links related by a link cobordism in kCP^2, originally conjectured by Manolescu-Marengon-Sarkar-Willis. As…

A 2-component link L is split if its components lie in disjoint balls. The boundary of either of these balls is called a splitting sphere for L. In the 3-sphere, 2-component split links have unique splitting spheres, meaning any two splitting spheres for L…

Using tools from lattice homology, we calculate the Seiberg-Witten Floer spectra of Seifert fibered spaces. We also discuss some speculative connections with algebraic geometry. This is joint work with Hirofumi Sasahira and Matthew Stoffregen.

In 1961, Mazur constructed a contractible, compact, smooth 4-manifold with boundary which is not homeomorphic to the standard 4-ball. In this talk, for any integer n ≥ 2 we construct a contractible, compact, smooth (n + 3)-manifold with boundary which is not homeomorphic to the…

Manolescu and Piccirillo introduced RBG links, a kind of 3-component framed links in S^3, that produce knot pairs with the same 0-surgery. In this talk, I will define n-RBG links, which generalizes their RBG construction to n-surgeries. I will explore the potential use of the s-invariant to…

We discuss new methods for using the Heegaard Floer homology
of hypersurfaces to distinguish between smooth closed 4-manifolds that
are homeomorphic but non-diffeomorphic. Specifically, for a 4-manifold X
with b_1(X)=1, the minimum rank of the reduced Heegaard Floer homology
of…

In joint work with Marco Marengon we present a simple but flexible
method to simultaneously remove multiple double points of immersed
surfaces in 4-​manifolds. One consequence is that in an appropriate
sense many knots bound disks in 4-​manifolds, and in particular, we
…

We will define a Bar-Natan homology for null homologous links in RP^3. As in the case for the usual Bar-Natan homology, this gives rise to a s-invariant and certain genus bound for null homologous knots in RP^3. More explicitly, it gives a genus bound for equivariant slice surface bounding the…

Khovanov homology extends to an invariant of smooth oriented 4-manifolds, which is defined as a skein module spanned by decorated embedded surfaces, modulo local relations. I will introduce equivariant and Lee versions of these skein modules and compute the latter. This leads to a non-vanishing…