The categorification of the Kauffman bracket skein module of $\mathbb{R}P^3$

TL;DR

This paper extends Khovanov homology to categorify the Kauffman bracket skein module for links in the twisted I-bundle over real projective plane, addressing previous limitations in non-orientable surface cases.

Contribution

It introduces a new differential in the Khovanov chain complex to successfully categorify the skein module in the non-orientable case of P^2,  imes I.

Findings

Successfully categorified the skein module for P^2 imes I

Redefined the differential in Khovanov chain complex

Extended homology theory to non-orientable surfaces

Abstract

Khovanov homology, an invariant of links in $\mathbb{R}^3$, is a graded homology theory that categorifies the Jones polynomial in the sense that the graded Euler characteristic of the homology is the Jones polynomial. Asaeda, Przytycki and Sikora generalized this construction by defining a double graded homology theory that categorifies the Kauffman bracket skein module of links in $I$-bundles over surfaces, except for the surface $\mathbb{R}P^2$, where the construction fails due to strange behaviour of links when projected to the non-orientable surface $\mathbb{R}P^2$. This paper categorifies the missing case of the twisted $I$-bundle over $\mathbb{R}P^2$, $\mathbb{R}P^2$ \widetilde{\times} I \approx \rpt \setminus \{\ast\}$, by redefining the differential in the Khovanov chain complex in a suitable manner.