Floer homology, low-dimensional topology, and algebra

Simons Collaboration Satellite Conference
McDonnell Hall A02

9:30 AM — 10:30 AM
Speaker: Maggie Miller, University of Texas at Austin
Location: McDonnell A02

Twisting, rolling, and branching in S^4 

In this talk, I’ll discuss two different constructions of “roll twist spun knots,” a family of knotted 2-spheres in the 4-sphere. One of these constructions is due to Litherland and the other essentially to Fintushel—Stern. The first is the standard construction, but the second makes it clear that certain cyclic branched covers of roll twist spun knots are diffeomorphic. As a consequence, we’ll see that a family of 4-manifolds recently constructed by Miyazawa are all diffeomorphic to the complex projective plane CP^2, meaning that his infinite family of exotic involutions are all defined on CP^2. This is joint work with Mark Hughes and Seungwon Kim.

10:45 AM — 11:45 AM
Speaker: Emmanuel Wagner, University of Paris
Location: McDonnell A02

Algebraic vs. geometric categorification of the Alexander polynomial: A spectral sequel

We construct a spectral sequence from the gl0-homology, an algebraic categorification of the Alexander polynomial to the knot Floer homology. This spectral sequence is of Bockstein type and comes from a subtle manipulation of coefficients. The main tools are quantum traces of foams and of singular Soergel bimodules. This is a joint work with Anna Beliakova, Kris Putyra, and Louis-Hadrien Robert.

1:30 PM — 2:30 PM
Speaker: Qiuyu Ren, UC Berkeley
Location: McDonnell A01

Khovanov skein lasagna module detects exotic 4-manifolds

We present new calculations of the Khovanov-Rozansky gl_2 skein lasagna modules defined by Morrison-Walker-Wedrich, generalizing several previous works. In particular, our calculation shows that the -1 traces on the knots -5_2 and P(3,-3,-8) have non-isomorphic skein lasagna modules, thus are non-diffeomorphic (while they are homeomorphic by Kirby moves + Freedman's result). This leads to the first gauge/Floer-theory-free proof of the existence of compact exotic 4-manifolds. Time permitting, we sketch some proofs or present some other results in our work. This is joint work with Michael Willis.

3:00 PM — 4:00 PM
Speaker: Sucharit Sarkar, UCLA
Location: McDonnell A01

A Link Floer spectrum via grid diagrams

Link Floer homology of links in S^3 can be computed as the homology of a grid chain complex defined using grid diagrams. I will (partially) describe a construction of a CW spectrum whose cells correspond to the generators of the grid chain complex, and whose cellular chain complex is the grid chain complex (and therefore, the homology is link Floer homology). This is joint with Ciprian Manolescu.