Hyperbolicity and identification of Berge knots of types VII and VIII

TL;DR

This paper proves that Berge knots of types VII and VIII are hyperbolic except for the known torus knots, using Reidemeister torsions and Alexander polynomials, and explores how standard parameters identify these knots.

Contribution

It establishes hyperbolicity for Berge knots of types VII and VIII beyond known cases and analyzes parameter-based identification methods.

Findings

Most Berge knots of types VII and VIII are hyperbolic.

Alexander polynomials confirm hyperbolicity.

Standard parameters can identify Berge knots of these types.

Abstract

T. Saito and M. Teragaito asked whether Berge knots of type VII are hyperbolic, and showed that some infinite sequences of the knots are hyperbolic. We show that Berge knots of types VII and VIII are hyperbolic except the known sequence of torus knots. We used the Reidemeister torsions. As a result, the Alexander polynomials of them have already shown their hyperbolicities. We also show that the standard parameters identify Berge knots of types VII and VIII, and study what kind of information identify them.