L-space knots with tunnel number >1 by experiment

TL;DR

This paper experimentally investigates L-space knots, revealing that among hyperbolic examples in Dunfield's catalog, 22 have tunnel number 2 and the rest have tunnel number 1, with detailed braid and genus properties.

Contribution

It provides the first systematic experimental classification of tunnel numbers for L-space knots in a hyperbolic census, including braid representations and asymmetry analysis.

Findings

22 hyperbolic L-space knots have tunnel number 2

Remaining L-space knots have tunnel number 1

Identified braid presentations satisfying Morton-Franks-Williams bound

Abstract

In Dunfield's catalog of the hyperbolic manifolds in the SnapPy census which are complements of L-space knots in $S^3$, we determine that $22$ have tunnel number $2$ while the remaining all have tunnel number $1$. Notably, these $22$ manifolds contain $9$ asymmetric L-space knot complements. Furthermore, using SnapPy and KLO we find presentations of these $22$ knots as closures of positive braids that realize the Morton-Franks-Williams bound on braid index. The smallest of these has genus $12$ and braid index $4$.