Computation of Large Asymptotics of 3-Manifold Quantum Invariants

TL;DR

This paper develops optimized algorithms to compute large quantum invariants of 3-manifolds, enabling experimental verification of conjectures and advancing computational topology methods.

Contribution

It introduces a new complexity parameter, combines preprocessing and multi-precision arithmetic, and improves computational efficiency for Turaev-Viro invariants.

Findings

Parameter accurately estimates enumeration complexity

Optimizations outperform existing implementations

Experimental verification of Chen and Yang's volume conjecture

Abstract

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.