Reidemeister torsion of two-bridge knots and signatures of TQFT

TL;DR

This paper links the Reidemeister torsion of two-bridge knots with SU₂-TQFT signatures, revealing invariance properties and asymptotic behaviors related to roots of unity and the Verlinde formula.

Contribution

It establishes an explicit relation between knot torsion and TQFT signatures, demonstrating invariance and asymptotic properties in these mathematical structures.

Findings

Inverse sum of torsions is constant, independent of p and q.

Signatures along certain roots of unity have asymptotic behavior matching the Verlinde formula.

The relation provides new insights into the interplay between knot invariants and quantum topology.

Abstract

We establish an explicit relation between the adjoint Reidemeister torsion of the two-bridge knot $K(p,q)$ at any parabolic representation and the Frobenius algebra governing the signatures of SU$_2$-TQFT vector spaces at the root $\zeta=\exp(i\pi q/p)$. As applications, (a) we prove that the inverse sum of torsions is constant (i.e., independent of $p$ and $q$); and (b) we show that along sequences of roots of the form $\zeta_n = \exp\left(i\pi\tfrac{a+bn}{c+dn}\right)$, the signatures have the same asymptotic behavior as the Verlinde formula.