Iterated splitting and the classification of knot tunnels

Details Iterated splitting and the classification of knot tunnels Iterated splitting and the classification of knot tunnels

2011-08-18

arxiv.org/abs/1108.3671v2

Mathematics

Geometric Topology

The results of the paper are unchanged. The list of known tunnel phenomena has been enlarged to include new possibilities seen

Scientific paper

For a genus-1 1-bridge knot in the 3-sphere, 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.

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