枚举基本法向曲面：算法、实验和不变量
Enumerating fundamental normal surfaces: Algorithms, experiments and invariants
摘要：
计算纽结理论和3-流形拓扑在最近几年已经看到了重大的突破，尽管许多关键算法的复杂性界限是指数或更大。在这种情况下，实验对于理解实用性的极限以及衡量竞争算法的相对优点至关重要。在本文中，我们专注于正常的表面理论，一个关键的工具，出现在整个低维拓扑。超越了计算顶点法向曲面（本质上是一个多面体圆锥的极端射线）的研究问题，我们把我们的注意力转移到计算基本法向曲面（本质上是这样一个圆锥的整基）的更复杂的任务。我们开发，实施和实验比较原始和对偶算法，这两个结合特定领域的技术与经典的希尔伯特基算法。我们的实验表明，我们可以解决曾经难以解决的非常大的问题。作为我们的技术的一个实际应用，我们通过计算398个以前未知的结的crosscap数来填补KnotInfo数据库中的空白。
英文摘要：
Computational knot theory and 3-manifold topology have seen significant breakthroughs in recent years, despite the fact that many key algorithms have complexity bounds that are exponential or greater. In this setting, experimentation is essential for understanding the limits of practicality, as well as for gauging the relative merits of competing algorithms. In this paper we focus on normal surface theory, a key tool that appears throughout low-dimensional topology. Stepping beyond the well-studied problem of computing vertex normal surfaces (essentially extreme rays of a polyhedral cone), we turn our attention to the more complex task of computing fundamental normal surfaces (essentially an integral basis for such a cone). We develop, implement and experimentally compare a primal and a dual algorithm, both of which combine domain-specific techniques with classical Hilbert basis algorithms. Our experiments indicate that we can solve extremely large problems that were once though intractable. As a practical application of our techniques, we fill gaps from the KnotInfo database by computing 398 previously-unknown crosscap numbers of knots.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
