Remarks on definition of Khovanov homology

TL;DR

This paper revisits Khovanov homology, providing a more topologically intuitive formulation and extending it to framed links, establishing relations via exact sequences for links with Kauffman brackets.

Contribution

It offers a topologically friendly reformulation of Khovanov homology and introduces a version for framed links, connecting homology groups through exact sequences.

Findings

Reformulation of Khovanov homology in topological terms

Introduction of Khovanov homology for framed links

Homology groups related by exact sequences for links with Kauffman brackets

Abstract

Mikhail Khovanov in math.QA/9908171 defined, for a diagram of an oriented classical link, a collection of groups numerated by pairs of integers. These groups were constructed as homology groups of certain chain complexes. The Euler characteristics of these complexes are coefficients of the Jones polynomial of the link. The goal of this note is to rewrite this construction in terms more friendly to topologists. A version of Khovanov homology for framed links is introduced. For framed links whose Kauffman brackets are involved in a skein relation, these homology groups are related by an exact sequence.