Random Walk on Knot Diagrams, Colored Jones Polynomial and Ihara-Selberg Zeta Function

Details Random Walk on Knot Diagrams, Colored Jones Polynomial and Ihara-Selberg Zeta Function Random Walk on Knot Diagrams, Colored Jones Polynomial and Ihara-Selberg Zeta Function

1998-12-07

arxiv.org/abs/math/9812039v1

Mathematics

Geometric Topology

18 pages

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Random Walk on Knot Diagrams, Colored Jones Polynomial and Ihara-Selberg Zeta Function does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Random Walk on Knot Diagrams, Colored Jones Polynomial and Ihara-Selberg Zeta Function, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Random Walk on Knot Diagrams, Colored Jones Polynomial and Ihara-Selberg Zeta Function will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-687898