New York Journal of Mathematics
Volume 29 (2023), 363-401


Khanh Le and Rebekah Palmer

Geodesic surfaces in the complement of knots with small crossing number

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Published: March 17, 2023.
Keywords: hyperbolic knots, totally geodesic surfaces.
Subject [2020]: 57K32.

In this article, we investigate the problem of counting totally geodesic surfaces in the complement of hyperbolic knots with at most 9 crossings. Adapting previous counting techniques of boundary slope and intersection, we establish uniqueness of a totally geodesic surface for the knots 74 and 935. Extending an obstruction to the existence of totally geodesic surfaces due to Calegari, we show that there is no totally geodesic surface in the complement of 47 knots.


The authors were supported by NSF Grant DMS 1906088. The authors would like to thank Nathan Dunfield for his suggestions concerning the computation in the case of the knot 925. The authors would like to thank Matthew Stover, Colin Adams, and Nathan Dunfield for their comments on an early draft of the paper. Finally, the authors would like to thank the anonymous referee for pointing out a gap in the proof of Proposition 4.4 in an earlier draft and for their comments in improving this paper.

Author information

Khanh Le
Department of Mathematics
Rice University
6100 Main Street
Houston, TX 77005, USA


Rebekah Palmer
Department of Mathematics
Temple University
1805 N Broad Street
Philadelphia, PA 19122, USA