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

## TL;DR

This paper develops a generalized skein relation for Khovanov homology using spectral sequences, enabling the categorification of the $ heta$-invariant, a broader generalization of the Jones polynomial.

## Contribution

It introduces a new skein-type relation for Khovanov homology and extends it to categorify the $ heta$-invariant, expanding the scope of link invariants.

## Key findings

- Derived a skein-type relation for Khovanov homology.
- Categorified the $ heta$-invariant using the new relation.
- Enhanced understanding of link invariants through homological methods.

## 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.