Journal of the Mathematical Society of Japan

Iterated splitting and the classification of knot tunnels

Sangbum CHO and Darryl MCCULLOUGH

Full-text: Open access

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.

Article information

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

Dates
First available in Project Euclid: 25 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1366896647

Digital Object Identifier
doi:10.2969/jmsj/06520671

Mathematical Reviews number (MathSciNet)
MR3055599

Zentralblatt MATH identifier
1270.57019

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

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

Citation

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. https://projecteuclid.org/euclid.jmsj/1366896647


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