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.

中文翻译：

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3-流形量子不变量的大渐近线计算

量子拓扑不变量在计算拓扑中发挥了重要作用，它们是现代主要数学猜想的核心。在本文中，我们研究了计算 Turaev-Viro 不变量 $\mathrm{TV}_r$ 的大 $r$ 值的实验问题。我们的方法基于优化的回溯算法，包括在 3 流形的三角剖分上枚举组合数据。我们设计了一个易于计算的参数来估计枚举空间的复杂性，基于多胞体中的格点计数，并通过实验证明其准确性。我们将此参数应用于三角剖分的预处理策略，并将其与多精度算法相结合，以计算 Turaev-Viro 不变量。