Residual Torsion-Free Nilpotence, Bi-Orderability and Pretzel Knots

TL;DR

This paper applies Mayland's technique to genus one and high-genus pretzel knots to identify many new examples of knots with bi-orderable groups, including non-fibered, non-alternating cases.

Contribution

It introduces new examples of bi-orderable knot groups by extending residual torsion-free nilpotence techniques to a broader class of pretzel knots.

Findings

Many new bi-orderable knot groups identified

First examples for non-fibered, non-alternating knots

Technique applicable to high-genus pretzel knots

Abstract

The residual torsion-free nilpotence of the commutator subgroup of a knot group has played a key role in studying the bi-orderability of knot groups. A technique developed by Mayland provides a sufficient condition for the commutator subgroup of a knot group to be residually-torsion-free nilpotent using work of Baumslag. In this paper, we apply Mayland's technique to several genus one pretzel knots and a family of pretzel knots with arbitrarily high genus. As a result, we obtain a large number of new examples of knots with bi-orderable knot groups. These are the first examples of bi-orderable knot groups for knots which are not fibered or alternating.