The adjoint Reidemeister torsion for the connected sum of knots

TL;DR

This paper extends the definition of the adjoint Reidemeister torsion to high-dimensional components of the character variety for connected sum knots and shows its local constancy and vanishing properties under certain conditions.

Contribution

It introduces a natural way to define the adjoint Reidemeister torsion for high-dimensional components of the character variety of connected sum knots and proves its key properties.

Findings

The adjoint Reidemeister torsion is well-defined on high-dimensional components.

It is locally constant where the meridian trace is fixed.

It satisfies a vanishing identity if each summand knot does.

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.