The adjoint Reidemeister torsion for the connected sum of knots

Abstract

Let K be the connected sum of knots K_1,\ldots,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 \geq 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.