Knot Groups with Many Killers

TL;DR

This paper demonstrates that certain knot groups, including those of torus and hyperbolic knots with specific properties, have infinitely many elements that normally generate the group and are not automorphic images of each other.

Contribution

It establishes the existence of infinitely many non-automorphic elements that normally generate the group in various classes of knot groups.

Findings

Existence of infinitely many normally generating elements in these knot groups.

These elements are pairwise non-automorphic.

The result applies to nontrivial torus, hyperbolic 2-bridge, and hyperbolic knots with unknotting number one.

Abstract

The group of any nontrivial torus knot, hyperbolic 2-bridge knot, or hyperbolic knot with unknotting number one contains infinitely many elements, none the automorphic image of another, such that each normally generates the group.