论文标题
计算三个manifolds中的基本表面
Counting essential surfaces in 3-manifolds
论文作者
论文摘要
我们考虑了3个序列中基本表面的同位同位素类别的自然问题，重点是在一系列双曲线3型脉络膜中封闭的基本表面。我们的主要结果是（可能是断开连接的）基本表面的计数，其特征总是具有短生成功能，因此具有准真主行为。这为此类表面的数量以及详细的渐近学提供了非常简洁的公式。我们提供算法，使我们能够计算这些生成功能和基础表面，并将其应用于近60,000个歧管，提供大量有关它们的数据。我们使用这些数据来探讨仅计算连接基本表面并提出一些猜想的精致问题。我们的方法涉及正常且几乎正常的表面，尤其是Tollefson和Oertel的工作，再加上Ehrhart开创的技术，用于计算Poliehedra中具有合理顶点的晶格点。我们还引入了一种新的测试方法，如果在理想的三角剖分中的正常表面是必不可少的，以避免切割沿表面的歧管。相反，我们在原始三角剖分中使用了几乎正常的表面。
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.
