Heegaard Floer homology, related invariants, and their applications

Abstract: We give an overview of some aspects of Heegaard Floer homology, with a focus on those that most directly relate to the group's talks.

Homology cobordism & smooth knot concordance

To a knot K in the 3-sphere we can associate the 3-manifold that arises from zero-framed Dehn surgery on K. It is a natural question to ask: if two knots have zero-surgeries which are Z-homology cobordant via cobordism W (with technical condition that the two positive knot meridians are homologous) does that imply that the knots must be concordant? In this talk, we give a pair of knots which are rationally concordant and whose zero-surgeries have an aforementioned homology cobordism between them, but the knots are not smoothly concordant. One knot in the pair is the figure eight knot, 4_1, which represents a 2-torsion element in the smooth concordance group C; all knots used in previous counterexamples represent infinite order elements. We will also discuss some related results involving iterating the pattern on 4_1 and other knots.

https://arxiv.org/abs/2210.10308

Maslov index formula in Heegaard Floer homology

Abstract: The formula introduced by Robert Lipshitz for Heegaard Floer homology is now one of the basic tools for those working with HF homology. The convenience of the formula is due to its combinatorial nature. In these talks, we will discuss the recent combinatorial proof of this formula. We will also talk more broadly about available combinatorial techniques, that one may use to work with arbitrary Heegaard diagrams.

https://arxiv.org/abs/2206.05221 

Knot lattice homotopy and the surgery formula

Abstract: In the first talk, I discuss how knot lattice homotopy relates to knot Floer homology and the broader Floer homotopy program. Along the way I discuss some of the challenges and opportunities that knot Floer homology, Floer homotopy, and specifically knot lattice homotopy provide. In my second talk, I discuss a surgery formula for knot lattice homotopy, which calculates the doubly filtered space associated to the dual knot post-surgery. Additionally the formula can, given the involutive data for the starting three-manifold and knot, reconstruct said data post-surgery. I put this in the context of other work on surgery formulas and sketch the steps needed to prove the surgery formula for knot lattice homotopy.

Unknotting number and (1,1) satellites

Abstract: The unknotting number of a knot, defined as the minimum number of crossing changes needed to untie it, is in general an intractable invariant. Its behavior under cabling is bounded below by recent work of Hom-Lidman-Park using knot Floer homology. I will discuss the unknotting number of satellite knots with (1,1)-patterns. This is joint work with Wenzhao Chen.

Naturality of Legendrian LOSS invariant under positive contact surgery and application

Abstract: We prove the naturality of LOSS invariant under all positive integer contact surgeries. Using the naturality we construction of new infinite families of examples of Legendrian non-simple knots that are distinguished by their LOSS invariants. In the first video we give necessary background and context and we show the result in the second video.

https://arxiv.org/abs/2306.09516