Framed knot contact homology
We extend knot contact homology to a theory over the ring ℤ[λ±1, μ±1] with the invariant given topologically and combinatorially. The improved invariant, which is defined for framed knots in S3 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.
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