Menasco–Reid conjecture on totally geodesic surfaces in hyperbolic knot complements

Let $S^{3}\setminus K$ be the complement of a hyperbolic knot $K$ in $S^{3}$. A closed embedded totally geodesic surface is a closed totally geodesic surface embedded in this hyperbolic $3$-manifold. Menasco–Reid conjecture. There does not exist a hyperbolic knot complement in $S^{3}$ that contains a closed embedded totally geodesic surface. Counterexamples are now known, although classifying the hyperbolic knots and links that fail to admit such surfaces remains an open and interesting question.