Framed knot contact homology

Details Framed knot contact homology Framed knot contact homology

2004-07-06

arxiv.org/abs/math/0407071v3

Duke Math. J. 141 (2008), no. 2, 365-406

Mathematics

Geometric Topology

v3: 36 pages, minor corrections, to appear in Duke Math. J

Scientific paper

We extend knot contact homology to a theory over the ring $\mathbb{Z}[\lambda^{\pm 1},\mu^{\pm 1}]$, with the invariant given topologically and combinatorially. The improved invariant, which is defined for framed knots in $S^3$ and can be generalized to knots in arbitrary manifolds, distinguishes the unknot and can distinguish mutants. It contains the Alexander polynomial and naturally produces a two-variable polynomial knot invariant which is related to the $A$-polynomial.

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