A conjecture of Chinburg-Reid-Stover for surgeries on twist knots

TL;DR

This paper investigates the set of prime numbers associated with quaternion algebras from Dehn surgeries on hyperbolic knots, providing evidence for a conjecture by Chinburg-Reid-Stover by showing infinitely many knots with an infinite set.

Contribution

It demonstrates the existence of infinitely many hyperbolic knots with an infinite set of associated prime numbers, advancing understanding of the Chinburg-Reid-Stover conjecture.

Findings

Existence of infinitely many knots with infinite prime sets

Counterexamples to finiteness conditions based on Alexander polynomial

Supports the conjecture of Chinburg-Reid-Stover

Abstract

Associated to a hyperbolic knot complement in $S^3$ is a set of prime numbers corresponding to the residue characteristics of the ramified places of the quaternion algebras obtained by Dehn surgery on the knots. Previous work by Chinburg-Reid-Stover gives conditions on the Alexander polynomial of the knot for this set to be finite. We show that there are infinitely many examples of knots for which this set is infinite, providing evidence for a conjecture of Chinburg-Reid-Stover.