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Hardness of Computation of Quantum Invariants on 3-Manifolds with Restricted Topology

Authors Henrique Ennes , Clément Maria

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ACM Subject Classification
  • Theory of computation → Computational geometry
  • Mathematics of computing → Geometric topology
  • Theory of computation → Problems, reductions and completeness
Keywords
  • 3-manifold
  • Heegaard splitting
  • Hempel distance
  • Quantum invariant
  • polynomial time reduction

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Abstract

Quantum invariants in low-dimensional topology offer a wide variety of valuable invariants about knots and 3-manifolds, presented by explicit formulas that are readily computable. Their computational complexity has been actively studied and is tightly connected to topological quantum computing. In this article, we prove that for any 3-manifold quantum invariant in the Reshetikhin-Turaev model, there is a deterministic polynomial time algorithm that, given as input an arbitrary closed 3-manifold M, outputs a closed 3-manifold M' with the same quantum invariant, such that M' is hyperbolic, contains no low genus embedded incompressible surface, and is presented by a strongly irreducible Heegaard diagram. Our construction relies on properties of Heegaard splittings and the Hempel distance. At the level of computational complexity, this proves that the hardness of computing a given quantum invariant of 3-manifolds is preserved even when severely restricting the topology and the combinatorics of the input. This positively answers a question raised by Samperton [Samperton, 2023].

Cite As Get BibTex

Henrique Ennes and Clément Maria. Hardness of Computation of Quantum Invariants on 3-Manifolds with Restricted Topology. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 37:1-37:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ESA.2025.37

Author Details

Henrique Ennes
  • Université Côte d'Azur, Centre INRIA, Sophia Antipolis, France
Clément Maria
  • Université Côte d'Azur, Centre INRIA, Sophia Antipolis, France

Funding

This work has been partially supported by the ANR project ANR-20-CE48-0007 (AlgoKnot) and the project ANR-15-IDEX-0001 (UCA JEDI). It has also been supported by the French government, through the France 2030 investment plan managed by the Agence Nationale de la Recherche, as part of the "UCA DS4H" project, reference ANR-17-EURE-0004.

Acknowledgements

We are grateful to Lukas Woike, Alex He, and Nicolas Nisse for helpful discussions on Vafa’s theorem, triangulations of Heegaard splittings, and general complexity theory, respectively. We also thank the anonymous reviewers for their thoughtful comments and constructive feedback, which greatly contributed to improving the clarity, rigor, and quality of this paper.

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