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THE CATEGORIFICATION OF THE KAUFFMAN BRACKET SKEIN MODULE OF
$ \mathbb{R} {\mathrm{P} }^{3} $
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 88 / Issue 3 / December 2013
- Published online by Cambridge University Press:
- 11 April 2013, pp. 407-422
- Print publication:
- December 2013
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- Article
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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.
