Stavros Garoufalidis
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 SU(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 C^2. We then compare it with the the character variety of SL(2,C) 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.
Key words: q-holonomic functions, D-modules, characteristic variety, deformation variety, colored Jones function, multisums, hypergeometric functions, WZ algorithm.
Notes: 13 pages, 3 figures.