On the characteristic and deformation varieties of a knot

Abstract

The colored Jones function of a knot is a sequence of Laurent polynomials in one variable, whose nth term is the Jones polynomial of the knot colored with the n-dimensional irreducible representation of sl 2 .It was recently shown by TTQ Le and the author that the colored Jones function of a knot is q-holonomic, i.e., that it satisfies a nontrivial linear recursion relation with appropriate coefficients.Using holonomicity, we introduce a geometric invariant of a knot: the characteristic variety, an affine 1-dimensional variety in 2 .We then compare it with the character variety of SL 2 ( ) representations, viewed from the boundary.The comparison is stated as a conjecture which we verify (by a direct computation) in the case of the trefoil and figure eight knots.We also propose a geometric relation between the peripheral subgroup of the knot group, and basic operators that act on the colored Jones function.We also define a noncommutative version (the so-called noncommutative A-polynomial) of the characteristic variety of a knot.Holonomicity works well for higher rank groups and goes beyond hyperbolic geometry, as we explain in the last chapter.