Generalized torsion for knots with arbitrarily high genus

Motegi, Kimihiko  ; Teragaito, Masakazu 

Canadian mathematical bulletin, Tome 65 (2022) no. 4, pp. 867-881

Résumé

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.

Détail

DOI : 10.4153/S0008439521000977

Mots-clés : Fundamental group, Dehn filling, slope, generalized torsion