Heegaard Floer homology, related invariants, and their applications
Organized by Fraser Binns and Subhankar Dey
Live Events
Reminder: All events are in Eastern (NYC/Boston/Atlanta) time. Registered participants will find Zoom links in the "Live Events" channel in the Discord server. If you aren't seeing any events below, scroll within the calendar and they will appear!
Introductory Talk: Fraser Binns and Subhankar Dey
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.
Talks (click the speaker name to see their title and abstract)
Holt Bodish
Three and Four dimensional invariants of satellite knots with (1,1) trefoil patterns
Abstract: In this talk, we discuss recent work computing knot Floer homology of a family of satellite knots with (1,1) trefoil patterns. We compute the three genus, bound the four genus, show that these patterns are fibered in the solid torus and show that satellites formed from these patterns with thin fibered companions have left or right veering monodromy.
Sally Collins
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.
Roman Krutowski
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.
Seppo Niemi-Colvin
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.
Weizhe Shen
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.
Shunyu Wan
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.