Pretzel Knots and q-Series

TL;DR

This paper explores the connection between the tail of the colored Jones polynomial for pretzel knots and false theta function identities, generalizing Ramanujan's identities and computing new examples.

Contribution

It introduces new q-series identities from pretzel knots' colored Jones polynomials, extending Ramanujan-type identities and providing explicit tail computations.

Findings

Proved a false theta function identity related to Ramanujan.

Generalized identities using pretzel knots' tails.

Computed tails for an infinite family of pretzel knots.

Abstract

The tail of the colored Jones polynomial of an alternating link is a $q$-series invariant whose first $n$ terms coincide with the first $n$ terms of the $n$-th colored Jones polynomial. Recently, it has been shown that the tail of the colored Jones polynomial of torus knots give rise to Ramanujan type identities. In this paper, we study $q$-series identities coming from the colored Jones polynomial of pretzel knots. We prove a false theta function identity that goes back to Ramanujan and we give a natural generalization of this identity using the tail of the colored Jones polynomial of Pretzel knots. Furthermore, we compute the tail for an infinite family of Pretzel knots and relate it to false theta function-type identities.