Tunnel-number-one knot exteriors in $S^3$ disjoint from proper power curves

Sungmo Kang

Vol. 315 (2021), No. 2, 369–418

DOI: 10.2140/pjm.2021.315.369

Abstract

As one of the background papers of the classification project of hyperbolic primitive/Seifert (or P/SF) knots in $S^3$ whose complete list is given in [Berge and Kang 2020], this paper classifies pairs of two disjoint nonseparating simple closed curves $R$ and $\beta$ lying in the boundary of a genus two handlebody $H$ such that $\beta$ is a proper power curve and a 2-handle addition $H\left[R\right]$ along $R$ embeds in $S^3$ so that $H\left[R\right]$ is the exterior of a tunnel-number-one knot. As a consequence, if $R$ is a nonseparating simple closed curve on the boundary of a genus two handlebody such that $H\left[R\right]$ embeds in $S^3$, then there exists a proper power curve disjoint from $R$ if and only if $H\left[R\right]$ is the exterior of the unknot, a torus knot, or a tunnel-number-one cable of a torus knot.

The results of this paper will be mainly used in proving the hyperbolicity of P/SF knots and in classifying P/SF knots in once-punctured tori in $S^3$, which is one of the types of P/SF knots in [Berge and Kang 2020]. Together with these results, the preliminary of this paper which consists of three parts: the three diagrams which are Heegaard diagrams, R-R diagrams, and hybrid diagrams, the “culling lemma”, and locating waves into an R-R diagrams, will also be used in the classification of hyperbolic primitive/Seifert knots in $S^3$.

Keywords

tunnel-number-one knots, primitive/Seifert knots, proper power curves, 2-handle additions, Heegaard and R-R diagrams

Mathematical Subject Classification

Primary: 57K10

Milestones

Received: 24 May 2020

Revised: 23 September 2021

Accepted: 17 October 2021

Published: 19 January 2022