论文标题
3个manifold量子不变的大渐近学计算
Computation of Large Asymptotics of 3-Manifold Quantum Invariants
论文作者
论文摘要
量子拓扑不变性在计算拓扑中发挥了重要作用，它们是主要的现代数学猜想的核心。在本文中，我们研究了计算turaev-viro不变性的大$ r $值的实验问题，$ \ mathrm {tv} _r $。我们将方法以优化的回溯算法为基础，该算法包括有关3个manifold的三角剖分的组合数据。我们设计了一个易于计算的参数，以根据多面体中的晶格计数估算枚举空间的复杂性，并通过实验表明其准确性。我们将此参数应用于三角剖分的预处理策略，并将其与多精液算术相结合，以计算Turaev-Viro不变性。我们最终研究了这些优化与最先进的实施相比所带来的改进，并在封闭的3个manifolds的人口普查中验证了实验性的陈和杨的体积猜想。
Quantum topological invariants have played an important role in computational topology, and they are at the heart of major modern mathematical conjectures. In this article, we study the experimental problem of computing large $r$ values of Turaev-Viro invariants $\mathrm{TV}_r$. We base our approach on an optimized backtracking algorithm, consisting of enumerating combinatorial data on a triangulation of a 3-manifold. We design an easily computable parameter to estimate the complexity of the enumeration space, based on lattice point counting in polytopes, and show experimentally its accuracy. We apply this parameter to a preprocessing strategy on the triangulation, and combine it with multi-precision arithmetics in order to compute the Turaev-Viro invariants. We finally study the improvements brought by these optimizations compared to state-of-the-art implementations, and verify experimentally Chen and Yang's volume conjecture on a census of closed 3-manifolds.
