2003/06/15 by Stavros Garoufalidis, Garoufalidis, Stavros
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Computational Geometry and Mesh Generation #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Quantum Algebra (math.QA)
paper · pdf · doi:10.48550/arxiv.math/0306229
openalex publication_date 2003/06/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The colored Jones function of a knot is a sequence of Laurent polynomials. It was shown by TTQ. Le and the author that such sequences are q-holonomic, that is, they satisfy linear q-difference equations with coefficients Laurent polynomials in q and qn. We show from first principles that q-holonomic sequences give rise to modules over a q-Weyl ring. Frohman-Gelca-LoFaro have identified the latter ring with the ring of even functions of the quantum torus, and with the Kauffman bracket skein module of the torus. Via this identification, we study relations among the orthogonal, peripheral and recursion ideal of the colored Jones function, introduced by the above mentioned authors. In the second part of the paper, we convert the linear q-difference equations of the colored Jones function in terms of a hierarchy of linear ordinary differential equations for its loop expansion. This conversion is a version of the WKB method, and may shed some information on the problem of asymptotics of the colored Jones function of a knot.Final version. To appear in the Journal of Knot Theory and its Ramifications.