Reidemeister torsion of two-bridge knots and signatures of TQFT

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.