THE CATEGORIFICATION OF THE KAUFFMAN BRACKET SKEIN MODULE OF RP3

Khovanov homology, an invariant of links in 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 et al. ['Categorification of the Kauffman bracket skein module of I-bundles over surfaces', Algebr. Geom. Topol. 4 (2004), 1177-1210] generalised 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 RP2, where the construction fails due to strange behaviour of links when projected to the nonorientable surface RP2. This paper categorifies the missing case of the twisted I-bundle over RP2, RP2(X) over tildeI approximate to RP3 \ {*}, by redefining the differential in the Khovanov chain complex in a suitable manner.