Homology of framed links embedded in thickened surfaces

TL;DR

This paper develops new homology theories for framed links in thickened surfaces, linking topological invariants to categorified algebraic structures, inspired by Khovanov homology and skein modules.

Contribution

It introduces an infinite family of homology theories for links in thickened surfaces, connecting them to the Kauffman bracket and surface embeddings, expanding categorification methods.

Findings

Constructed an infinite family of homology theories.

One theory's Euler characteristic equals the Kauffman bracket.

Uses embedded surfaces to generate chain groups.

Abstract

We construct an infinite family of homology theories of framed links in thickened surfaces, as well as a homology theory whose graded Euler characteristic is exactly the Kauffman bracket of the link in the surface. Both theories are based on ideas coming from Asaeda, Przytycki and Sikora's categorification of the Kauffman bracket skein module of I-bundles over surfaces. This is accomplished by borrowing ideas from Bar-Natan's Khovanov homology theory for tangles and cobordisms and using embedded surfaces to generate the chain groups, instead of diagrams.