Step by step categorification of the Jones polynomial in Kauffman's version

TL;DR

This paper develops a step-by-step categorification of the Jones polynomial in Kauffman's version, constructing new link invariants through homology theories that extend and relate to Khovanov homology.

Contribution

It introduces a novel categorification process for the Jones polynomial in Kauffman's framework, including a new homology invariant for oriented links.

Findings

Constructed a 2-complex on Kauffman's states cube that yields a link invariant homology.

Extended the homology to incorporate orientation, producing a new invariant homology for oriented links.

Clarified the relationship between this new homology and the original Khovanov homology.

Abstract

Given any diagram of a link, we define on the cube of Kauffman's states a "2-complex" whose homology is an invariant of the associated framed links, and such that the graded Euler characteristic reproduces the unnormalized Kauffman bracket. This includes a categorification of brackets skein relation. Then we incorporate the orientation information and get a further complex on the same cube that gives rise to a new invariant homology for oriented links, so that the graded Euler characteristic reproduces the unnormalized Jones polynomial in Kauffman's version. Finally we clarify the relations between this homology and the original Khovanov homology of oriented links, extending the well known relation between the associated two versions of the Jones polynomial.