Floer homology, low-dimensional topology, and algebra

Simons Collaboration Satellite Conference
Fine Hall 110

9:00 AM — 9:55 AM
Speaker: Ian Zemke, Princeton University
Location: Fine Hall 110

On L-space links

A well-known result of Ozsvath and Szabo describes the knot Floer complex of an L-space knot in terms of its Alexander polynomial. The family of L-space knots includes all algebraic knots. The situation of L-space links is comparably less understood. We will describe a conjectural description of the link Floer complex, as well as proof of this conjecture for plumbed L-space links (this family includes all algebraic links) as well as 2-component L-space links. The case of plumbed L-space links is joint with M. Borodzik and B. Liu, and the case of 2-component L-space links is joint work in progress with D. 

Chen and H. Zhou. In addition to describing the conjecture, we will study several instructive examples. The case of 2-component L-space links has applications towards computations of certain satellite operators in Heegaard Floer theory.

10:00 AM — 10:55 AM
Speaker: Andy Manion, NC State University
Location: Fine Hall 110

Spectral arc algebras from the perspective of skew Howe duality

I will talk about recent joint work with Anne Dranowski, Meng Guo, and Aaron Lauda toward defining a gl_2-foam version of Lawson-Lipshitz-Sarkar's spectrified Khovanov arc algebra H^n, in line with the general expectations of skew Howe duality. I will also mention a perspective on spectrified H^n that uses certain canonical multimerge cobordisms called "frames."

3:00 PM — 4:00 PM
Speaker: Jonathan Hanselman, Princeton University
Location: McDonnell A01

Satellite knots and immersed curves

Satellite operations are a valuable method of constructing complicated knots from simpler ones, and much work has gone into understanding how various 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 diagram obtained from a Heegaard diagram for the pattern and the immersed curve representing the UV=0 knot Floer complex of the companion.

This is particularly useful for (1,1)-patterns, since in this case the resulting immersed diagram is genus one and the computation is combinatorial. In the case of one-bridge braid satellites the immersed curve invariant for the satellite can be obtained directly from that of the companion by deforming the diagram, generalizing earlier work with Watson on cables. This is joint work with Wenzhao Chen.