$Q$-Series and Quantum Spin Networks

TL;DR

This paper investigates the tail functions of quantum spin networks, computes their $q$-series, and explores their relation to the colored Jones polynomial, revealing a natural product structure in these networks.

Contribution

It introduces the existence and computation of tail functions for quantum spin networks and establishes their relation to the colored Jones polynomial, highlighting a natural product structure.

Findings

Computed $q$-series for an infinite family of networks

Established relation between network tails and colored Jones polynomial

Demonstrated a natural product structure in the studied networks

Abstract

The tail of a quantum spin network in the two-sphere is a $q$-series associated to the network. We study the existence of the head and tail functions of quantum spin networks colored by $2n$. We compute the $q$-series for an infinite family of quantum spin networks and give the relation between the tail of these networks and the tail of the colored Jones polynomial. Finally, we show that the family of quantum spin networks under study satisfies a natural product structure, making these networks satisfy a natural product structure.