1998/12/07 by Xiao-Song Lin, Zhenghan Wang, Lin, Xiao-Song +1
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #FOS: Mathematics #Geometric Topology (math.GT) #Probability (math.PR) #Quantum Algebra (math.QA) #Stochastic processes and statistical mechanics #math.GT #math.PR #math.QA
paper · pdf · doi:10.48550/arxiv.math/9812039
18 pages
arxiv created 1998/12/07 · openalex publication_date 1998/12/07 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A model of random walk on knot diagrams is used to study the Alexander polynomial and the colored Jones polynomial of knots. In this context, the inverse of the Alexander polynomial of a knot plays the role of an Ihara-Selberg zeta function of a directed weighted graph, counting with weights cycles of random walk on a 1-string link whose closure is the knot in question. The colored Jones polynomial then counts with weights families of ``self-avoiding'' cycles of random walk on the cabling of the 1-string link. As a consequence of such interpretations of the Alexander and colored Jones polynomials, the computation of the limit of the renormalized colored Jones polynomial when the coloring (or cabling) parameter tends to infinity whereas the weight parameter tends to 1 leads immediately to a new proof of the Melvin-Morton conjecture, which was first established by Rozansky and by Bar-Natan and Garoufalidis.