L-space knots with tunnel number >1 by experiment

ABSTRACT

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.

KEYWORDS:

• Braid
• L-space knot
• asymmetric
• SnapPy

2010 MATHEMATICS SUBJECT CLASSIFICATION::

• Primary 57M25
• 57M27
• Secondary 57R58

Acknowledgements

This article began as a record of the results of an “office hour” session on the use of SnapPy [12] and KLO [33] held at the ICERM workshop Perspectives on Dehn Surgery, July 15–19, 2019. We thank ICERM (the Institute for Computational and Experimental Research in Mathematics in Providence, RI) for the productive environment where this work could be carried out and Nathan Dunfield for his interest and assistance. Furthermore, we thank to all of SnapPy’s creators/contributors and KLO’s Frank Swenton for producing and maintaining these ever-useful programs. We also thank to the anonymous referees for useful suggestions and the recommendation to expand the scope of this work.

Declaration of Interest

No potential conflict of interest was reported by the author(s).

Notes

1 Personal communication.

2 These two unclassified knots have now been shown to be actually L-space knots [4].

3 Recent work shows that none of the knots in $\mathcal{A}$ admit an alternating surgery [4].

4 See also https://snappy.math.uic.edu/installing.html#kitchen-sink

5 This problem will appear on a problem list compiled from the ICERM workshop Perspectives on Dehn Surgery.

Additional information

Funding

As a part of the ICERM workshop, this work is supported in part by the National Science Foundation under Grant No. DMS-1439786 and by the NSF CAREER Award DMS-1455132. KLB was partially supported by a grant from the Simons Foundation (grant #523883 to Kenneth L. Baker). KM was partially supported by NSF RTG grant DMS-1344991 and by the Simons Foundation. SO was partially supported by Turkish Academy of Sciences TÜBA-GEBİP Award. ST’s attendance at the workshop was supported by the Dartmouth Department of Mathematics. AW’s attendance at the workshop was supported by the grant NSF Career Award DMS-1455132.