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

Abstract: We consider a natural divisor on $\operatorname{SL}_2(\mathbb C)$-character varieties of knots and links, given by the so-called acyclic Reidemeister torsion. We provide a geometric interpretation of this divisor. We focus on the particular family of odd twisted Whitehead links, where we show that this divisor has multiplicity two. Moreover, we apply these results to the study of the $L$-functions of the universal deformations of representations over finite fields of twisted Whitehead links.