Bundles of coloured posets and a Leray-Serre spectral sequence for Khovanov homology

TL;DR

This paper develops a bundle theory for coloured posets, introducing a spectral sequence that converges to Khovanov homology, thereby providing new computational tools in link homology theory.

Contribution

It introduces a novel bundle theory for coloured posets and constructs a Leray-Serre type spectral sequence applicable to Khovanov homology.

Findings

Established a spectral sequence for coloured posets.

Applied the spectral sequence to compute Khovanov homology.

Provided new insights into the structure of Khovanov homology.

Abstract

The decorated hypercube found in the construction of Khovanov homology for links is an example of a Boolean lattice equipped with a presheaf of modules. One can place this in a wider setting as an example of a coloured poset, that is to say a poset with a unique maximal element equipped with a presheaf of modules. In this paper we initiate the study of a bundle theory for coloured posets, producing for a certain class of base posets a Leray-Serre type spectral sequence. We then show how this theory finds application in Khovanov homology by producing a new spectral sequence converging to the Khovanov homology of a given link.