Geodesic surfaces in the complement of knots with small crossing number

TL;DR

This paper studies the existence and uniqueness of totally geodesic surfaces in the complements of hyperbolic knots with up to 9 crossings, providing new classifications and obstructions.

Contribution

It introduces adapted counting techniques for boundary slopes and intersection, establishing uniqueness for certain knots and proving non-existence for others.

Findings

Unique totally geodesic surface for knots 7_4 and 9_35

No totally geodesic surfaces in 47 knot complements

Extended obstruction criteria for geodesic surfaces

Abstract

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 $7_4$ and $9_{35}$. 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.