Incompressible planar surfaces in hyperbolic link exteriors in the 3-sphere

Abstract

For each integer $N\ge 3$ we construct examples of $N$-component hyperbolic links $L\subset {\mathbb{S}}^3$ whose exterior contains an incompressible spanning planar surface $P\subset X_L$ with one boundary component on each boundary torus of $X_L$ of nonmeridional and nonintegral slope, thus providing counterexamples to a recent conjecture of M. Eudave-Muñoz and M. Ozawa. The case $N=3$ is the crucial one to consider: all such link pairs $(L,P)$ are classified and found to be generated by the structure of the exterior of hyperbolic Eudave-Muñoz knots. More generally, necessary and sufficient conditions on integers $p_1,p_2,p_3\ge 2$ are given for the existence of a 3-component link in ${\mathbb{S}}^3$ whose exterior contains a spanning pants with boundary slopes of the form $a_i/p_i$. A key role in the analysis of 3-component link pairs is played by the properties of the embeddings of three mutually disjoint and nonparallel primitive circles on the boundary of a genus two handlebody. These are classified in general and in the special case when the handlebody is part of a genus two Heegaard decomposition of ${\mathbb{S}}^3$ associated with a 3-component link pair. The hyperbolic links with $N\ge 4$ components whose exterior contains a spanning planar surface with nonmeridional and nonintegral boundary slopes are constructed via an inductive process that starts with any of the classified 3-component hyperbolic link pairs.