Coxeter Quotients of Knot Groups through 16 Crossings

Abstract

We find explicit maximal rank Coxeter quotients for the knot groups of $595,515$ out of the $1,701,936$ knots through 16 crossings. We thus calculate the bridge numbers and verify Cappell and Shaneson’s Meridional Rank Conjecture for these knots. In addition, we provide a computational tool for establishing the conjecture for those knots beyond 16 crossings whose meridional ranks can be detected via finite Coxeter quotients.

KEYWORDS:

• Knot
• Coxeter group
• bridge number
• meridional rank

2010 AMS SUBJECT CLASSIFICATION:

• 57M25
• 57M27
• 57M05

Acknowledgments

The authors would like to thank Curtis Bennett, John Brevik and Jon McCammond for helpful conversations about Coxeter groups; and Sebastian Baader and Levi Ryffel for offering feedback on a draft of this paper. A portion of this work was completed while AK was a guest at the Max Planck Institute for Mathematics in Bonn. We are grateful to MPIM for its hospitality.

Notes

1 Again, the surface is assumed to be reduced, which means that each disk is incident to at least 3 bands (otherwise the disk becomes absorbed into a band) and the crossings in each band have the same sign.

Additional information

Funding

RB and NM were partially supported by NSF grant DMS-1821254. AK was partially supported by NSF grant DMS-2204349.