Geodesic surfaces in the complement of knots with small crossing number

2022/07/29 by Khanh Le, Le, Khanh, Rebekah Palmer +1
Mathematics · #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2207.14792

openalex publication_date 2022/07/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 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.

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