A Polynomial-Time Algorithm to Compute Turaev–Viro Invariants \(\mathrm {TV}_{4,q}\) of 3-Manifolds with Bounded First Betti Number


In this article, we introduce a fixed-parameter tractable algorithm for computing the Turaev–Viro invariants \(\mathrm {TV}_{4,q}\), using the first Betti number, i.e. the dimension of the first homology group of the manifold with \(\mathbb {Z}_2\)-coefficients, as parameter. This is, to our knowledge, the first parameterised algorithm in computational 3-manifold topology using a topological parameter. The computation of \(\mathrm {TV}_{4,q}\) is known to be #P-hard in general; using a topological parameter provides an algorithm polynomial in the size of the input triangulation for the family of 3-manifolds with first \(\mathbb {Z}_2\)-homology group of bounded dimension. Our algorithm is easy to implement, and running times are comparable with running times to compute integral homology groups for standard libraries of triangulated 3-manifolds. The invariants we can compute this way are powerful: in combination with integral homology and using standard data sets, we are able to almost double the pairs of 3-manifolds we can distinguish. We hope this qualifies \(\mathrm {TV}_{4,q}\) to be added to the short list of standard properties (such as orientability, connectedness and Betti numbers) that can be computed ad hoc when first investigating an unknown triangulation.

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  1. 1.

    Having a one-vertex triangulation is a rather weak restriction; see Sect. 4.

  2. 2.

    Integral homology groups, i.e. homology groups with integer coefficients, are strictly more powerful than homology groups with finite field coefficients, but can still be computed in polynomial time; see [37, Chapter 8] for an overview of algorithms.

  3. 3.

    Most notably, Turaev and Viro do not consider triangle weights \(|f|_\theta \), but instead incorporate an additional factor of \(|f|_\theta ^{1/2}\) into each tetrahedron weight \(|t|_\theta \) and \(|t'|_\theta \) for the two tetrahedra t and \(t'\) containing f.

  4. 4.

    See [18] for an alternative surface interpretation of admissible colourings for small r.

  5. 5.

    The procedure may fail in the rare case of triangulations containing two-sided projective planes. In such a case, the algorithm terminates in polynomial time stating the existence of such a surface.

  6. 6.

    In case \(\mathfrak {T}\) contains a two-sided projective plane and the algorithm fails, this fact is detected by the algorithm.

  7. 7.

    Note that computing \(\mathrm {TV}_{4,q}\) is not a counting problem and, as such, not a #P problem in nature.


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We would like to thank the anonymous referees for numerous helpful comments on an earlier version of this article. This work is supported by the Australian Research Council (projects DP140104246 and DP150104108).

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Correspondence to Jonathan Spreer.

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An extended abstract [25] of this article has been published in the Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA) 2017.

Communicated by Herbert Edelsbrunner.

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Maria, C., Spreer, J. A Polynomial-Time Algorithm to Compute Turaev–Viro Invariants \(\mathrm {TV}_{4,q}\) of 3-Manifolds with Bounded First Betti Number. Found Comput Math 20, 1013–1034 (2020). https://doi.org/10.1007/s10208-019-09438-8

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  • Fixed-parameter tractable algorithms
  • Turaev–Viro invariants
  • Triangulations of 3-manifolds
  • (Integral)homology
  • (Generalised)normal surfaces
  • Discrete algorithms

Mathematics Subject Classification

  • 57M27
  • 57Q15
  • 68Q25