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

  • Clément Maria
  • Jonathan SpreerEmail author


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.


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

Mathematics Subject Classification

57M27 57Q15 68Q25 



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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© SFoCM 2019

Authors and Affiliations

  1. 1.INRIA Sophia-Antipolis-MéditerranéeValbonneFrance
  2. 2.The University of SydneySydneyAustralia

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