Categorification of the Colored Jones Polynomial and Rasmussen Invariant of Links

By Anna Beliakova, S. Wehrli. Canadian Journal of Mathematics - Journal Canadien de Mathematiques, 2005

95 citations recorded by the source corpus.

Abstract 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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/32d33c67f4c1d49cd7cfd96ddfa2a73ba28999f5

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Citations in the corpus

Categorifying fractional Euler characteristics, Jones–Wenzl projectors and 3j-symbols (2012, background): “On the other hand, our approach gives rise to a different (and actually much more subtle) colored Jones invariant than the original construction of Khovanov in [36] and the one in [13].”
Categorification of the Jones-Wenzl Projectors (2010, methodology): “It is important to note that our definition disagrees with some previous categorifications based on different axiomatizations of the Jones–Wenzl projectors such as the dimension axiom [17] (for related work see [5] and [11]): Tr.”
Man and machine thinking about the smooth 4-dimensional Poincaré conjecture (2009, background): “Kh[Knot[8, 19], Program→ ”FastKh”, Modulus→ 5]”