The adjoint Reidemeister torsion for the connected sum of knots

Abstract

Let $K$ be the connected sum of knots $K_1,\dots ,K_n$. It is known that the ${\mathrm{SL}}_2(\mathbb{C})$-character variety of the knot exterior of $K$ has a component of dimension $\ge 2$ as the connected sum admits a so-called bending. We show that there is a natural way to define the adjoint Reidemeister torsion for such a high-dimensional component and prove that it is locally constant on a subset of the character variety where the trace of a meridian is constant. We also prove that the adjoint Reidemeister torsion of $K$ satisfies the vanishing identity if each $K_i$ does so.