Semisimple tunnels

TL;DR

This paper develops a method to translate between braid group elements and slope invariants of tunnels in genus-1 1-bridge knots, enabling calculations and characterizations of tunnel invariants for various knot classes.

Contribution

It introduces an algorithm to compute slope invariants from braid descriptions and characterizes the slope sequences for 2-bridge and toroidal tunnels.

Findings

Verified slope invariants for all 2-bridge and (1,1)-tunnels of torus knots.

Developed and implemented software for calculating slope invariants.

Provided characterizations of slope sequences for specific tunnel classes.

Abstract

A knot in the 3-sphere 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.