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THE CATEGORIFICATION OF THE KAUFFMAN BRACKET SKEIN MODULE OF $ \mathbb{R} {\mathrm{P} }^{3} $

Part of: Low-dimensional topology Differential topology

Published online by Cambridge University Press:  11 April 2013

BOŠTJAN GABROVŠEK
BOŠTJAN GABROVŠEK*
Affiliation:
Faculty of Mathematics and Physics, University of Ljubljana, Jadranska ulica 19, SI-1000 Ljubljana, Slovenia email [email protected]
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Abstract

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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.

Keywords

Khovanov homologycategorificationskein moduleKauffman bracket

MSC classification

Secondary: 57M27: Invariants of knots and 3-manifolds 57M25: Knots and links in $S^3$ 57R56: Topological quantum field theories
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 88 , Issue 3 , December 2013 , pp. 407 - 422
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

References

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