Generalized torsion for knots with arbitrarily high genus

TL;DR

This paper introduces a new class of hyperbolic knots with generalized torsion, including knots of arbitrarily high genus, and explores their properties related to bi-orderability of knot groups.

Contribution

It demonstrates the existence of infinitely many hyperbolic knots with generalized torsion via p-twisting, extending the class of known knots with this property.

Findings

Infinite family of hyperbolic knots with generalized torsion

Knots with arbitrarily high genus exhibit generalized torsion

Some twisted torus knots, including the (-2, 3, 7)-pretzel knot, have generalized torsion

Abstract

In a group, a non-trivial element is called a generalized torsion element if some non-empty finite product of its conjugates equals to the identity. We say that a knot has generalized torsion if its knot group admits such an element. For a (2, 2q+1)-torus knot K, we demonstrate that there are infinitely many unknots c such that p-twisting K about c yields a twist family, which consists of hyperbolic knots with generalized torsion whenever |p| > 3. This gives a new infinite class of hyperbolic knots having generalized torsion. In particular, each class contains knots with arbitrarily high genus. We also show that some twisted torus knots, including the (-2, 3, 7)-pretzel knot, have generalized torsion. Since generalized torsion is an obstruction for having bi-order, these knots have non-bi-orderable knot groups.