From state integrals to q-series

TL;DR

This paper demonstrates how multi-dimensional state integrals in Quantum Topology can be expressed as finite sums of basic hypergeometric series, providing detailed proofs for specific knot invariants.

Contribution

It offers a detailed proof linking state integrals of quantum topology to q-series, expanding understanding of their algebraic structure.

Findings

State integrals can be written as finite sums of hypergeometric series.

Explicit proofs provided for invariants of 4_1 and 5_2 knots.

Bridges between quantum topology and q-series established.

Abstract

It is well-known to the experts that multi-dimensional state integrals of products of Faddeev's quantum dilogarithm which arise in Quantum Topology can be written as finite sums of products of basic hypergeometric series in q=e^{2\pi i\tau} and \tilde{q}=e^{-2\pi i/\tau}. We illustrate this fact by giving a detailed proof for a family of one-dimensional integrals which includes state-integral invariants of 4_1 and 5_2 knots.