Counting essential surfaces in 3-manifolds

• 成果类型: Article
• 署名作者: Dunfield, Nathan M.; Garoufalidis, Stavros; Rubinstein, J. Hyam
• 署名单位: University of Illinois System; University of Illinois Urbana-Champaign; Southern University of Science & Technology; University of Melbourne
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-021-01090-w
• 发表日期: 2022
• 页码: 717-775
• 关键词: closed incompressible surfaces; rational generating-functions; measured laminations; triangulations; knot; algorithm; degenerations; points; genus

摘要

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.