Geodesic surfaces in the complement of knots with small crossing number

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.

arXiv comment29 pages, 7 figures. Comments are welcome. Updated with corrected typos and filled in gap in the proof of Proposition 4.4. To appear in the New York Journal of Math