L-space knots with tunnel number >1 by experiment

In Dunfield's catalog of the hyperbolic manifolds in the SnapPy census which are complements of L-space knots in S, 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.