Framed knot contact homology

Abstract

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 that is related to the $A$-polynomial