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In this article, we investigate the problem of counting totally geodesicsurfaces in the complement of hyperbolic knots with at most 9 crossings.Adapting previous counting techniques of boundary slope and intersection, weestablish uniqueness of a totally geodesic surface for the knots $7_4$ and$9_{35}$. Extending an obstruction to the existence of totally geodesicsurfaces due to Calegari, we show that there is no totally geodesic surface inthe complement of 47 knots.

Geodesic surfaces in the complement of knots with small crossing number