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.