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There are no unexpected tunnel number one knots of genus one

Author(s): Martin Scharlemann
Journal: Trans. Amer. Math. Soc. 356 (2004), 1385-1442.
MSC (2000): Primary 57M25
Posted: October 6, 2003
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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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Additional Information:

Martin Scharlemann
Affiliation: Department of Mathematics, University of California, Santa Barbara, California 93106
Email: mgscharl@math.ucsb.edu

DOI: 10.1090/S0002-9947-03-03182-9
PII: S 0002-9947(03)03182-9
Received by editor(s): July 24, 2001
Received by editor(s) in revised form: July 25, 2002
Posted: October 6, 2003
Additional Notes: This research was supported in part by an NSF grant, the Miller Institute, and RIMS Kyoto
Copyright of article: Copyright 2003, American Mathematical Society

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