Curved Rickard complexes and link homologies

TL;DR

This paper introduces curved Rickard complexes, a deformation of categorified quantum group complexes, enabling new link homology invariants that generalize previous models and connect algebraic and geometric categorifications.

Contribution

It defines curved Rickard complexes and constructs deformed link homologies that extend existing theories to broader representations and partitions.

Findings

Deformed link homologies generalize previous models.

Connection established between algebraic and geometric categorifications.

New braid group actions via curved Rickard complexes.

Abstract

Rickard complexes in the context of categorified quantum groups can be used to construct braid group actions. We define and study certain natural deformations of these complexes which we call curved Rickard complexes. One application is to obtain deformations of link homologies which generalize those of Batson-Seed arXiv:1303.6240 and Gorsky-Hogancamp arXiv:1712.03938 to arbitrary representations/partitions. Another is to relate the deformed homology defined algebro-geometrically in arXiv:1410.7156 to categorified quantum groups (this was the original motivation for this paper).