Iterated splitting and the classification of knot tunnels

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 nontorus-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.

Middle tunnels by splitting

“…For this, one needs the slopes of the drop-and lift-disks. In fact, there is a general slope calculation that covers all four cases (as well as additional cases that will arise in [7]). In this section, we present this general slope calculation, and in the next section, we present the calculation of the slopes of disks γ n .…”

“…For the slope invariant, we set up a general method in Sections 5 and 6. Besides adding the transparency of abstraction, the setup will be used in [7] to calculate the slope invariants obtained by an iteration of the splitting construction, which we will discuss momentarily. Section 7 uses the general method to give the slopes in all cases of the splitting construction, and Section 8 illustrates them for the Goda-Hayashi-Ishihara example.…”

“…We mentioned a further generalization of the splitting construction. In [7], we show how one can start with a tunnel obtained by a splitting construction and carry out an iteration of similar constructions, producing a much larger class of knots having both regular and semisimple tunnels. Each of the four splitting constructions admits two kinds of iteration sequences, giving eight versions of the iterated construction.…”