Counting essential surfaces in 3-manifolds

TL;DR

This paper develops a method to count isotopy classes of essential surfaces in hyperbolic 3-manifolds, revealing that their enumeration follows a quasi-polynomial pattern and providing algorithms to compute these counts across thousands of manifolds.

Contribution

It introduces a new approach to count essential surfaces using normal and almost normal surfaces, with algorithms and conjectures for connected surfaces.

Findings

Counting functions have short generating functions with quasi-polynomial behavior.

Algorithms successfully computed data for nearly 60,000 manifolds.

New criteria for testing essentiality of normal surfaces without cutting the manifold.

Abstract

We consider the natural problem of counting isotopy classes of essential surfaces in 3-manifolds, focusing on closed essential surfaces in a broad class of hyperbolic 3-manifolds. Our main result is that the count of (possibly disconnected) essential surfaces in terms of their Euler characteristic always has a short generating function and hence has quasi-polynomial behavior. This gives remarkably concise formulae for the number of such surfaces, as well as detailed asymptotics. We give algorithms that allow us to compute these generating functions and the underlying surfaces, and apply these to almost 60,000 manifolds, providing a wealth of data about them. We use this data to explore the delicate question of counting only connected essential surfaces and propose some conjectures. Our methods involve normal and almost normal surfaces, especially the work of Tollefson and Oertel, combined with techniques pioneered by Ehrhart for counting lattice points in polyhedra with rational vertices. We also introduce a new way of testing if a normal surface in an ideal triangulation is essential that avoids cutting the manifold open along the surface; rather, we use almost normal surfaces in the original triangulation.