Annular Khovanov-Lee homology, braids, and cobordisms

TL;DR

This paper develops an invariant for links in a thickened annulus using Khovanov-Lee homology, providing new tools to analyze braid properties and cobordisms with potential applications in knot theory.

Contribution

It introduces a bifiltered complex structure for Khovanov-Lee homology in the annular setting and defines annular Rasmussen invariants to study braid and cobordism properties.

Findings

Provides a necessary condition for braid quasipositivity.

Offers a sufficient condition for right-veeringness.

Establishes invariance of the bifiltered complex under isotopy in the annulus.

Abstract

We prove that the Khovanov-Lee complex of an oriented link, L, in a thickened annulus, A x I, has the structure of a bifiltered complex whose filtered chain homotopy type is an invariant of the isotopy class of L in A x I. Using ideas of Ozsvath-Stipsicz-Szabo as reinterpreted by Livingston, we use this structure to define a family of annular Rasmussen invariants that yield information about annular and non-annular cobordisms. Focusing on the special case of annular links obtained as braid closures, we use the behavior of the annular Rasmussen invariants to obtain a necessary condition for braid quasipositivity and a sufficient condition for right-veeringness.