On the characteristic and deformation varieties of a knot
Stavros Garoufalidis
TL;DR
The paper proposes a deep link between quantum knot invariants and classical geometric structures by showing that the colored Jones function is $q$-holonomic, which yields a characteristic variety that, conjecturally, matches the boundary SL2(C) deformation variety of the knot. It develops a polynomial framework via the $A$-polynomial and a noncommutative counterpart, $A_q$, and demonstrates how to compute $A_q$ through multisums and the Wilf–Zeilberger algorithm, with explicit verification for the trefoil and figure-eight knots. The results connect quantum recursions to classical character varieties and introduce higher-rank generalizations, suggesting a broad, algorithmic bridge between quantum invariants and geometric topology. The practical computations for 3_1 and 4_1 provide concrete evidence and computational methodologies (WZ, qZeil, etc.) for deriving and verifying these correspondences, offering tools for exploring the conjectures in broader knot families. Together, these ideas advance the program of understanding how quantum knot invariants encode the geometry of knot complements and their representations.
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.