Multiplicity of non-acyclic ${\rm SL}_2$-representations and L-functions of the odd-twisted Whitehead links

Abstract

We study the divisor of the Reidemeister torsion on the variety of irreducible ${\rm SL}_2\mathbb{C}$-characters of certain knots and links, and provide a geometric interpretation of them. We focus in particular on the family of odd-twisted Whitehead links $W_{2n-1}$ and prove that these divisors have multiplicity two. Furthermore, we apply these results to the study of the $L$-functions of the universal deformations of representations over fields with characteristic $p>2$ of these link groups.