#### Vol. 288, No. 2, 2017

 Recent Issues Vol. 293: 1 Vol. 292: 1  2 Vol. 291: 1  2 Vol. 290: 1  2 Vol. 289: 1  2 Vol. 288: 1  2 Vol. 287: 1  2 Vol. 286: 1  2 Online Archive Volume: Issue:
 The Journal Subscriptions Editorial Board Officers Special Issues Submission Guidelines Submission Form Contacts Author Index To Appear ISSN: 0030-8730
Knots of tunnel number one and meridional tori

### Mario Eudave-Muñoz and Grissel Santiago-González

Vol. 288 (2017), No. 2, 319–342
##### 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
Primary: 57M25
##### Milestones
Revised: 7 August 2016
Accepted: 19 September 2016
Published: 28 April 2017
##### Authors
 Mario Eudave-Muñoz Instituto de Matemáticas Universidad Nacional Autónoma de México Circuito Exterior Ciudad Universitaria 04510 México DF Mexico CIMAT Callejón Jalisco s/n Col. Valenciana 36240 Guanajuato Mexico Grissel Santiago-González Facultad de Ciencias Universidad Nacional Autónoma de México Circuito Exterior Ciudad Universitaria 04510 México, DF Mexico