Semisimple tunnels

Abstract

A knot in S 3 in genus-1 1-bridge position (called a (1, 1)-position) can be described by an element of the braid group of two points in the torus. Our main results tell how to translate between a braid group element and the sequence of slope invariants of the upper and lower tunnels of the (1, 1)-position. After using them to verify previous calculations of the slope invariants for all tunnels of 2-bridge knots and (1, 1)-tunnels of torus knots, we obtain characterizations of the slope sequences of tunnels of 2-bridge knots, and of a class of tunnels we call toroidal. The main results lead to a general algorithm to calculate the slope invariants of the upper and lower tunnels from a braid description. The algorithm has been implemented as software, and we give some sample computations.

Iterated splitting and the classification of knot tunnels

“…In this section, we will see that any drop-ρ iteration of the first kind examined in Sections 2 and 3 and starting with the trivial knot positioned as T 1,1 produces a 2-bridge knot in the (1, 1)-position whose upper tunnel is the upper semisimple tunnel of the knot, and moreover that every semisimple tunnel of every 2-bridge knot can be obtained in this way. We wil use the notation and the description of the classification of 2bridge knots presented in [8,Section 10]. We first recall the calculation of the slope invariants of the upper semisimple tunnel of a 2-bridge knot given in [8,Proposition 10.4]: fraction [2a d , 2b d , . .…”

“…We wil use the notation and the description of the classification of 2bridge knots presented in [8,Section 10]. We first recall the calculation of the slope invariants of the upper semisimple tunnel of a 2-bridge knot given in [8,Proposition 10.4]: fraction [2a d , 2b d , . . .…”

“…We have not included a review of the general theory of [4], as condensed reviews are already available in several of our articles. For the present paper, we surmise that Section 1 of [6] together with the review sections of [8] form the best option for most readers.…”

Middle tunnels by splitting

“…Each tunnel obtained by the splitting construction is associated to a (1, 1)position of its associated knot, and in Section 9 we explain how the method of [6] allows an easy calculation of the slope invariants of its upper and lower tunnels. As usual, we apply these to the Goda-Hayashi-Ishihara example.…”

“…We have not included a review of the general theory, as the original theory is detailed in [2] and brief reviews are already available in several of our articles. For the present paper, we would guess that Section 1 of [4] together with the review sections of [6] form the best option for most readers. Figure 2 shows a standard Heegaard torus T in S 3 , and an oriented longitude-meridian pair {ℓ, m} which will be our ordered basis for H 1 (T ) and for the homology of a product neighborhood T × I.…”