A generalized skein relation for Khovanov homology and a categorification of the $\theta$-invariant

Abstract

The Jones polynomial is a famous link invariant that can be defined diagrammatically via a skein relation. Khovanov homology is a richer link invariant that categorifies the Jones polynomial. Using spectral sequences, we obtain a skein-type relation satisfied by the Khovanov homology. Thanks to this relation, we are able to generalize the Khovanov homology in order to obtain a categorification of the $\theta$-invariant, which is itself a generalization of the Jones polynomial.