Knots of tunnel number one and meridional tori

Mario Eudave-Muñoz and Grissel Santiago-González

Vol. 288 (2017), No. 2, 319–342

DOI: 10.2140/pjm.2017.288.319

Abstract

We give a characterization of knots of tunnel number 1 that admit an essential meridional torus with two boundary components. Let $K$ be a knot in $S^3$, $S$ an essential meridional torus in the exterior of $K$ with two boundary components, and $\tau$ an unknotting tunnel for $K$. We consider the intersections between $S$ and $\tau$. If the intersection is empty, we conclude that the knot $K$ is an iterate of a satellite knot of tunnel number 1 and one of its unknotting tunnels, and then $S$ is knotted as a nontrivial torus knot. If the intersection is nonempty, we simplify it as much as possible, and conclude that the knot $K$ is a $\left(1,1\right)$-knot; it follows from known results that in some cases the torus $S$ is knotted as a nontrivial torus knot, while in others cases the torus $S$ is  unknotted.

Keywords

knot of tunnel number one, $(1,1)$-knot, meridional torus, iterate knot

Mathematical Subject Classification 2010

Primary: 57M25

Milestones

Received: 1 July 2015

Revised: 7 August 2016

Accepted: 19 September 2016

Published: 28 April 2017