# Finiteness of the image of the Reidemeister torsion of a splice

## TL;DR

This paper proves that the set of Reidemeister torsion values for certain spliced 3-manifolds is finite, using character varieties and A-polynomials, contributing to understanding torsion invariants in 3-manifold topology.

## Contribution

It establishes finiteness of Reidemeister torsion values for spliced knot complements, linking it to character varieties and A-polynomials, which was previously unknown.

## Key findings

- RT(M) is finite for certain knot splices
- Character varieties influence torsion finiteness
- A-polynomials relate to torsion set properties

## Abstract

The set $\mathit{RT}(M)$ of values of the $\mathit{SL}(2,\mathbb{C})$-Reidemeister torsion of a 3-manifold $M$ can be both finite and infinite. We prove that $\mathit{RT}(M)$ is a finite set if $M$ is the splice of two certain knots in the 3-sphere. The proof is based on an observation on the character varieties and $A$-polynomials of knots.