Iterated splitting and the classification of knot tunnels
arXiv:1108.3671
Abstract
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.
The results of the paper are unchanged. The list of known tunnel phenomena has been enlarged to include new possibilities seen in examples recently found by John Berge, after reading the previous version of the paper. The previous list was presented as a conjecture of all possibilities, but the new list is presented only as list of known phenomena, prompting the change of title