Generalized torsion for knots with arbitrarily high genus

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 $(2, 2q+1)$ -toru...

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Veröffentlicht in:Canadian mathematical bulletin 2022-12, Vol.65 (4), p.867-881
Hauptverfasser: Motegi, Kimihiko, Teragaito, Masakazu
Format: Artikel
Sprache:eng
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Zusammenfassung: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.
ISSN:0008-4395
1496-4287
DOI:10.4153/S0008439521000977