Foundations of the Colored Jones Polynomial of singular knots

TL;DR

This paper develops the foundational theory for the colored Jones polynomial of singular knots, extending existing algorithms and introducing new stability properties that connect to classical identities like Ramanujan's false theta function.

Contribution

It extends Masbum and Vogel's algorithm to compute the colored Jones polynomial for singular knots and introduces the tail and stability properties of these polynomials.

Findings

Extended algorithm for singular knots

Defined the tail of the colored Jones polynomial

Proved a Ramanujan false theta function identity

Abstract

This article gives the foundations of the colored Jones polynomial for singular knots. We extend Masbum and Vogel's algorithm to compute the colored Jones polynomial for any singular knot. We also introduce the tail of the colored Jones polynomial of singular knots and use its stability properties to prove a false theta function identity that goes back to Ramanujan.