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<p>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 <i>M</i>, outputs a closed 3-manifold <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(M'\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>M</mi> <mo>′</mo> </msup> </math></EquationSource> </InlineEquation> with the same quantum invariant, such that <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(M'\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>M</mi> <mo>′</mo> </msup> </math></EquationSource> </InlineEquation> 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&#xa0;(Samperton 2023).</p>

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Hardness of computation of quantum invariants on 3-manifolds with restricted topology

  • 期刊论文

  • 发表日期：

    2026年6月25日

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  • Henrique Ennes,
  • Clément Maria

摘要

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 \(M'\) M ′ with the same quantum invariant, such that \(M'\) 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).