Residual torsion-free nilpotence, biorderability and pretzel knots

Johnson, Jonathan

doi:10.2140/agt.2023.23.1787

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Residual torsion-free nilpotence, biorderability and pretzel knots

Jonathan Johnson

Algebraic & Geometric Topology 23 (2023) 1787–1830

DOI: 10.2140/agt.2023.23.1787

Abstract

The residual torsion-free nilpotence of the commutator subgroup of a knot group has played a key role in studying the biorderability of knot groups. A technique developed by Mayland (1975) provides a sufficient condition for the commutator subgroup of a knot group to be residually torsion-free nilpotent using work of Baumslag (1967, 1969). 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 biorderable knot groups. These are the first examples of biorderable knot groups for knots which are not fibered or alternating.

Keywords

pretzel knots, biorderable, free factor, commutator subgroups of pretzel knots

Mathematical Subject Classification

Primary: 57K10

References

Publication

Received: 13 September 2020

Revised: 3 November 2021

Accepted: 7 December 2021

Published: 14 June 2023

Authors

Jonathan Johnson
Department of Mathematics University of Texas at Austin Austin, TX United States

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Volume 23, issue 4 (2023)