Geodesic surfaces in the complement of knots with small crossing number

Khanh Le; Rebekah Palmer

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arXiv · 2207.14792

Geodesic surfaces in the complement of knots with small crossing number

Khanh Le·Rebekah Palmer

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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.

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Khanh Le, Rebekah Palmer. 2022-07-29. Geodesic surfaces in the complement of knots with small crossing number. https://arxiv.org/abs/2207.14792

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