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Generalized torsion for knots with arbitrarily high genus
Published online by Cambridge University Press: 02 December 2021
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.
MSC classification
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- Information
- Copyright
- © Canadian Mathematical Society, 2021
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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