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There are no unexpected tunnel number one knots of genus one. (English) Zbl 1042.57003
In this paper the author proves a conjecture first made by Goda and Teragaito [H. Goda and M. Teragaito, Tokyo J. Math. 22, No. 1, 99–103 (1999; Zbl 0939.57009)] by showing 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 M. Eudave-Munoz [Topology Appl. 55, No. 2, 131–152 (1994; Zbl 0816.57014)] and by K. Morimoto and M. Sakuma [Math. Ann. 289, No. 1, 143–167 (1991; Zbl 0697.57002)]. Recall that a knot in the 3-sphere is said to be tunnel number one if there exists an arc attached to the knot at its endpoints so that the complement of a regular neighborhood of the resulting complex is a genus two handlebody. Let \(\gamma\) be an unknotting tunnel for a knot \(K\). In [M. Scharlemann and A. Thompson, Proc. Lond. Math. Soc., III. Ser. 87, No. 2, 523–544 (2003; Zbl 1047.57008)] the authors proved that if \(\gamma\) can be slid and isotoped to lie on a genus one Seifert surface then \(K\) is necessarily a \(2\)-bridge knot. If not, \(\gamma\) can be slid and isotoped to an unknotted loop (this is the hard part of the paper; it uses thin position). Then \(K\) admits a \((1,1)\) decomposition (that is it is \(1\)-bridge on an unknotted torus) and for this class of knots H. Matsuda [Proc. Am. Math. Soc. 130, No. 7, 2155–2163 (2002; Zbl 0996.57004)] has proved the conjecture.

MSC:
57M25 Knots and links in the \(3\)-sphere (MSC2010)
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References:
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