#### Vol. 315, No. 2, 2021

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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
Primary: 57K10