The adjoint Reidemeister torsion for the connected sum of knots

Abstract

Let K be the connected sum of knots K 1 , . . . , Kn. It is known that the SL 2 (C)-character variety of the knot exterior of K has a component of dimension ≥ 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.

Twisted Reidemeister torsions via Heegaard splittings

“…Then, it is conjectured [1,3,4] that the sum lies in Z and, that if M is hyperbolic and n = −1, then the sum is zero. This conjecture is sometimes called the vanishing identity ; see [11,12,16] and references therein for supporting evidence of this conjecture.…”