Annular Khovanov-Lee homology, Braids, and Cobordisms

Elisenda Grigsby (Boston College)

Monday 13th March, 2017 16:00-17:00 Maths 204

Abstract

Khovanov homology associates to a knot K in the three-sphere a bigraded vector space arising as the homology groups of an abstract chain complex. Using a deformation of Khovanov's complex, due to Lee, Rasmussen defined an integer-valued knot invariant he called s(K) that gives a lower bound on the 4-ball genus of knots, sharp for knots that can be realized as quasipositive braid closures.

On the other hand, when K is a braid closure, its Khovanov complex can itself be realized in a natural way as a deformation of a triply-graded complex, defined by Asaeda-Przytycki-Sikora, further studied by L. Roberts, and now known as the (sutured) annular Khovanov complex.

In this talk, I will describe joint work with Tony Licata and Stephan Wehrli aimed at understanding an annular version of Lee's deformation of the Khovanov complex. In particular, we obtain a family of real-valued braid conjugacy class invariants generalizing Rasmussen's "s" invariant that give bounds on the Euler characteristic of smoothly-imbedded surfaces in the thickened solid torus as well as information about the associated mapping class of the punctured disk. The algebraic model for this construction is the Upsilon invariant of Ozsvath-Stipsicz-Szabo.

 

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