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$ \mathbb{R} {\mathrm{P} }^{3} $Published online by Cambridge University Press: 11 April 2013
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 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 
$ \mathbb{R} {\mathrm{P} }^{2} $, where the construction fails due to strange behaviour of links when projected to the nonorientable surface 
$ \mathbb{R} {\mathrm{P} }^{2} $. This paper categorifies the missing case of the twisted 
$I$-bundle over 
$ \mathbb{R} {\mathrm{P} }^{2} $, 
$ \mathbb{R} {\mathrm{P} }^{2} \widetilde {\times } I\approx \mathbb{R} {\mathrm{P} }^{3} \setminus \{ \ast \} $, by redefining the differential in the Khovanov chain complex in a suitable manner.
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