Vol. 288, No. 2, 2017

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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 S3 , S an essential meridional torus in the exterior of K with two boundary components, and τ an unknotting tunnel for K. We consider the intersections between S and τ. 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 (1,1)-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
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