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Generalized torsion for knots with arbitrarily high genus

Part of: Low-dimensional topology

Published online by Cambridge University Press:  02 December 2021

Kimihiko Motegi Open the ORCID record for Kimihiko Motegi [Opens in a new window]  and
Masakazu Teragaito Open the ORCID record for Masakazu Teragaito [Opens in a new window]
Kimihiko Motegi
Affiliation:
Department of Mathematics, Nihon University, 3-25-40 Sakurajosui, Setagaya-ku, Tokyo, 156-8550, Japan e-mail: [email protected]
Masakazu Teragaito*
Affiliation:
Department of Mathematics and Mathematics Education, Hiroshima University, 1-1-1 Kagamiyama, Higashi-hiroshima, Hiroshima, 739-8524, Japan
*
e-mail: [email protected]
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Abstract

Let G be a group, and let g be a nontrivial element in G. If some nonempty finite product of conjugates of g equals the identity, then g is called a generalized torsion element. We say that a knot K has generalized torsion if $G(K) = \pi _1(S^3 - K)$ admits such an element. For a $(2, 2q+1)$ -torus knot K, we demonstrate that there are infinitely many unknots $c_n$ in $S^3$ such that p-twisting K about $c_n$ yields a twist family $\{ K_{q, n, p}\}_{p \in \mathbb {Z}}$ in which $K_{q, n, p}$ is a hyperbolic knot 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. Because generalized torsion is an obstruction for having bi-order, these knots have non-bi-orderable knot groups.

Keywords

Fundamental groupDehn fillingslopegeneralized torsion

MSC classification

Secondary: 57M05: Fundamental group, presentations, free differential calculus 57M50: Geometric structures on low-dimensional manifolds
Type
Article
Information
Canadian Mathematical Bulletin , Volume 65 , Issue 4 , December 2022 , pp. 867 - 881
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Copyright
© Canadian Mathematical Society, 2021

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Footnotes

Dedicated to the memory of Toshie Takata. The first-named author has been partially supported by JSPS KAKENHI Grant Number 19K03502 and Joint Research Grant of the Institute of Natural Sciences at Nihon University for 2021. The second-named author has been supported by JSPS KAKENHI Grant Number 20K03587.

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