Journal of the Mathematical Society of Japan

Iterated splitting and the classification of knot tunnels

Sangbum CHO and Darryl MCCULLOUGH

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For a genus-1 1-bridge knot in $S^3$, that is, a (1,1)-knot, a middle tunnel is a tunnel that is not an upper or lower tunnel for some (1,1)-position. Most torus knots have a middle tunnel, and non-torus-knot examples were obtained by Goda, Hayashi, and Ishihara. In a previous paper, we generalized their construction and calculated the slope invariants for the resulting examples. We give an iterated version of the construction that produces many more examples, and calculate their slope invariants. If one starts with the trivial knot, the iterated constructions produce all the 2-bridge knots, giving a new calculation of the slope invariants of their tunnels. In the final section we compile a list of the known possibilities for the set of tunnels of a given tunnel number 1 knot.

Article information

J. Math. Soc. Japan, Volume 65, Number 2 (2013), 671-686.

First available in Project Euclid: 25 April 2013

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Zentralblatt MATH identifier

Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}

knot tunnel (1,1) torus knot regular splitting 2-bridge


CHO, Sangbum; MCCULLOUGH, Darryl. Iterated splitting and the classification of knot tunnels. J. Math. Soc. Japan 65 (2013), no. 2, 671--686. doi:10.2969/jmsj/06520671.

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