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Rasmussen's invariant and surfaces in some four-manifolds

Banff International Research Station for Mathematical Innovation and Discovery

Back in 2004, Rasmussen extracted a numerical invariant from Khovanov-Lee homology, and used it to give a new proof of Milnor's conjecture about the slice genus of torus knots. In this talk, I will describe a generalization of Rasmussen's invariant to null-homologous links in connected sums of $S^1 \times S^2$. For certain links in $S^1 \times S^2$, we compute the invariant by reinterpreting it in terms of Hochschild homology. As applications, we prove inequalities relating the Rasmussen-type invariant to the genus of null-homologous surfaces with boundary in the following four-manifolds: $B^2 \times S^2, S^1 \times B^3, CP^2$, and various connected sums and boundary sums of these. We deduce that Rasmussen's invariant also gives genus bounds for surfaces inside homotopy 4-balls obtained from $B^4$ by Gluck twists. Therefore, it cannot be used to prove that such homotopy 4-balls are non-standard. This is based on joint work with Marco Marengon, Sucharit Sarkar, and Mike Willis.

Mathematics; Nonassociative rings and algebras; Category theory; homological algebra; Representation theory

video/mp4

Author affiliation: UCLA

2020-06-01

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/74618

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/