There are no unexpected tunnel number one knots of genus one

Scharlemann, Martin

doi:10.1090/S0002-9947-03-03182-9

There are no unexpected tunnel number one knots of genus one
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Trans. Amer. Math. Soc. 356 (2004), 1385-1442 Request permission

Abstract:

We show that the only knots that are tunnel number one and genus one are those that are already known: $2$-bridge knots obtained by plumbing together two unknotted annuli and the satellite examples classified by Eudave-Muñoz and by Morimoto and Sakuma. This confirms a conjecture first made by Goda and Teragaito.

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