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

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.