#### Vol. 258, No. 1, 2012

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Semisimple tunnels

### Sangbum Cho and Darryl McCullough

Vol. 258 (2012), No. 1, 51–89
##### Abstract

A knot in ${S}^{3}$ in genus-$1$ $1$-bridge position (called a $\left(1,1\right)$-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 $\left(1,1\right)$-position. After using them to verify previous calculations of the slope invariants for all tunnels of $2$-bridge knots and $\left(1,1\right)$-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.

A knot in ${S}^{3}$ in genus-$1$ $1$-bridge position (called a $\left(1,1\right)$-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 $\left(1,1\right)$-position. After using them to verify previous calculations of the slope invariants for all tunnels of $2$-bridge knots and $\left(1,1\right)$-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.

A knot in ${S}^{3}$ in genus-$1$ $1$-bridge position (called a $\left(1,1\right)$-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 $\left(1,1\right)$-position. After using them to verify previous calculations of the slope invariants for all tunnels of $2$-bridge knots and $\left(1,1\right)$-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.

A knot in ${S}^{3}$ in genus-$1$ $1$-bridge position (called a $\left(1,1\right)$-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 $\left(1,1\right)$-position. After using them to verify previous calculations of the slope invariants for all tunnels of $2$-bridge knots and $\left(1,1\right)$-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.

A knot in ${S}^{3}$ in genus-$1$ $1$-bridge position (called a $\left(1,1\right)$-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 $\left(1,1\right)$-position. After using them to verify previous calculations of the slope invariants for all tunnels of $2$-bridge knots and $\left(1,1\right)$-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.

A knot in ${S}^{3}$ in genus-$1$ $1$-bridge position (called a $\left(1,1\right)$-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 $\left(1,1\right)$-position. After using them to verify previous calculations of the slope invariants for all tunnels of $2$-bridge knots and $\left(1,1\right)$-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.

##### Keywords
knot, tunnel, $(1,1)$, braid, torus, slope, invariant, cabling, semisimple, 2-bridge, toroidal, algorithm
Primary: 57M25
##### Supplementary material

Zip file of Python code for algorithms in the article