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April, 2013 Iterated splitting and the classification of knot tunnels
Sangbum CHO, Darryl MCCULLOUGH
J. Math. Soc. Japan 65(2): 671-686 (April, 2013). DOI: 10.2969/jmsj/06520671

Abstract

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.

Citation

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Sangbum CHO. Darryl MCCULLOUGH. "Iterated splitting and the classification of knot tunnels." J. Math. Soc. Japan 65 (2) 671 - 686, April, 2013. https://doi.org/10.2969/jmsj/06520671

Information

Published: April, 2013
First available in Project Euclid: 25 April 2013

zbMATH: 1270.57019
MathSciNet: MR3055599
Digital Object Identifier: 10.2969/jmsj/06520671

Subjects:
Primary: 57M25

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

Rights: Copyright © 2013 Mathematical Society of Japan

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Vol.65 • No. 2 • April, 2013
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