The complete $10$-tetrahedra census of orientable cusped hyperbolic $3$-manifolds
Shana Yunsheng Li
TL;DR
This work extends the census of orientable cusped hyperbolic $3$-manifolds to the case of $10$ tetrahedra, delivering a comprehensive catalog of $150{,}730$ manifolds and $496{,}638$ minimal ideal triangulations. It leverages triangulation theory, normal-surface techniques, and geometric criteria to certify minimality and realizability, aided by computational tools such as SnapPy. The authors report $439{,}898$ exceptional Dehn fillings and identify $1{,}849$ of the simplest hyperbolic knot exteriors in $S^3$, along with the simplest known example of a cusped orientable hyperbolic $3$-manifold containing a closed totally geodesic surface. They establish that all cusped hyperbolic $3$-manifolds triangulable by at most $10$ tetrahedra admit geometric triangulations and provide open data and code for reproducibility at the $10$-tetrahedra resource. This census significantly augments the landscape of hyperbolic $3$-manifold examples and supports systematic exploration of Dehn fillability and knot exteriors within a large, computable dataset.
Abstract
We extend the complete census of orientable cusped hyperbolic $3$-manifolds to $10$ tetrahedra, giving the next $150730$ manifolds and their $496638$ minimal ideal triangulations. As applications, we find the precisely $439898$ exceptional Dehn fillings on them, revealing the next $1849$ simplest hyperbolic knot exteriors in $S^3$. We also give the simplest example of an orientable cusped hyperbolic $3$-manifold containing a closed totally geodesic surface.