Quantum Knot Invariant for Torus Link and Modular Forms

TL;DR

This paper explores the asymptotic behavior of quantum invariants for torus links, connecting them to modular forms and Eichler integrals through q-series identities and limits at roots of unity.

Contribution

It introduces a new perspective linking quantum invariants of torus links to modular forms via q-series identities and asymptotic analysis.

Findings

Invariant expressed as a limit of Eichler integral at roots of unity

Established q-series identities related to torus link invariants

Connected quantum invariants with modular forms of weight 3/2

Abstract

We consider an asymptotic expansion of Kashaev's invariant or the colored Jones function for the torus link T(2,2m). We shall give q-series identity related to these invariants, and show that the invariant is regarded as a limit of q being N-th root of unity of the Eichler integral of the modular form of weight 3/2.