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Categorification of the Colored Jones Polynomial and Rasmussen Invariant of Links

Published online by Cambridge University Press:  20 November 2018

Anna Beliakova  and
Stephan Wehrli
Anna Beliakova
Affiliation:
Institut für Mathematik, Universität Zürich, CH-8057 Zürich, Switzerland e-mail:anna@math.unizh.ch, stephan.wehrli@math.unizh.ch
Stephan Wehrli
Affiliation:
Institut für Mathematik, Universität Zürich, CH-8057 Zürich, Switzerland e-mail:anna@math.unizh.ch, stephan.wehrli@math.unizh.ch
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Abstract

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We define a family of formal Khovanov brackets of a colored link depending on two parameters. The isomorphism classes of these brackets are invariants of framed colored links. The Bar-Natan functors applied to these brackets produce Khovanov and Lee homology theories categorifying the colored Jones polynomial. Further, we study conditions under which framed colored link cobordisms induce chain transformations between our formal brackets. We conjecture that for special choice of parameters, Khovanov and Lee homology theories of colored links are functorial (up to sign). Finally, we extend the Rasmussen invariant to links and give examples where this invariant is a stronger obstruction to sliceness than the multivariable Levine–Tristram signature.

Keywords

Primary: 57M25secondary: 57M2718G60Khovanov homologycolored Jones polynomialslice genusmovie movesframed cobordism
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 60 , Issue 6 , 01 December 2008 , pp. 1240 - 1266
Copyright
Copyright © Canadian Mathematical Society 2008

References

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