The colored Jones function is q-holonomic

Abstract

A function of several variables is called holonomic if, roughly speaking, it is determined from finitely many of its values via finitely many linear recursion relations with polynomial coefficients. Zeilberger was the first to notice that the abstract notion of holonomicity can be applied to verify, in a systematic and computerized way, combinatorial identities among special functions. Using a general state sum definition of the colored Jones function of a link in 3-space, we prove from first principles that the colored Jones function is a multisum of a q -proper-hypergeometric function, and thus it is q -holonomic. We demonstrate our results by computer calculations.

Asymptotics of the colored Jones function of a knot

“…In [GL05] , the authors gave state-sum formulas for J g,K similar to (28) where the summand takes values in Z[q ±1/D ], where D is the size of the center of g. The methods of the present paper give an upper bound for the growth-rate of the g-colored Jones function. More precisely, we have:…”

“…In fact, one may obtain an independent proof of Theorem 1.6 using WKB analysis, that is, the study of asymptotics of solutions of difference equations with a small parameter. The key idea is that the sequence of colored Jones functions is a solution of a linear q-difference equation, as was established in [GL05]. A discussion on WKB analysis of q-difference equations was given by Geronimo and the first author in [GG06].…”

Link polynomial calculus and the AENV conjecture

“…After getting these explicit formulas, we converted them into a differential expansion form a la [25,36]. As usual, such formulas in symmetric representations have a pronounced q-hypergeometric form (in accordance with a generic statement of [45]) and are easily continued to arbitrary values of r, s and t.…”

Three-dimensional extensions of the Alday-Gaiotto-Tachikawa relation

“…As a manifestation of this hidden structure, the averages < K > R satisfy K-dependent difference equations in the R(!) variable [26], which allows one to consider them as belonging to the family of generalized q-hypergeometric series. A q → 1 limit of these equations defines the spectral curve Σ(K) and the saddle point of the corresponding integral representation defines the associated Seiberg-Witten (SW) differential.…”

“…In [26] the polynomial invariants (13) for the gauge group SU (2) with spin J were systematically interpreted as generalized q-hypergeometric functions, such representations were widely used before in particular examples. This means that these Wilson averages can be represented in the form of finite sums:…”