On the characteristic and deformation varieties of a knot

TL;DR

This paper introduces the characteristic variety of a knot derived from the colored Jones function, compares it with the character variety, and explores its geometric and algebraic properties, including a noncommutative version.

Contribution

It defines the characteristic variety from the colored Jones function, compares it with the character variety, and proposes a noncommutative A-polynomial, advancing the understanding of knot invariants.

Findings

Verified the conjecture for trefoil and figure-eight knots.

Established a geometric relation between the knot group and colored Jones operators.

Introduced a noncommutative A-polynomial of a knot.

Abstract

The colored Jones function of a knot is a sequence of Laurent polynomials in one variable, whose n-th 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, ie, 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 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.