Rogers-Ramanujan type identities and the head and tail of the colored Jones polynomial

TL;DR

This paper investigates the head and tail of the colored Jones polynomial for alternating links, revealing their dependence on reduced checkerboard graphs and establishing a monoid structure for these functions.

Contribution

It introduces new combinatorial identities for the head and tail functions and shows their dependence solely on reduced checkerboard graphs, expanding understanding of their algebraic structure.

Findings

Head and tail functions depend only on reduced checkerboard graphs.

The class of head and tail functions forms a monoid for prime alternating links.

New combinatorial identities relate to Rogers-Ramanujan type identities.

Abstract

We study the head and tail of the colored Jones polynomial while focusing mainly on alternating links. Various ways to compute the colored Jones polynomial for a given link give rise to combinatorial identities for those power series. We further show that the head and tail functions only depend on the reduced checkerboard graphs of the knot diagram. Moreover the class of head and tail functions of prime alternating links forms a monoid.