Advertisement

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
Article
  • 26 Downloads

Abstract

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.

Keywords

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

Mathematics Subject Classification

57M27 57Q15 68Q25 

Notes

Acknowledgements

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).

References

  1. 1.
    Arnborg, S., Corneil, D.G., Proskurowski, A.: Complexity of finding embeddings in a \(k\)-tree. SIAM J. Algebraic Discrete Methods 8(2), 277–284 (1987)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bodlaender, H.L.: A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput. 25(6), 1305–1317 (1996)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Burton, B.A.: Structures of small closed non-orientable 3-manifold triangulations. J. Knot Theory Ramifications 16(5), 545–574 (2007)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Burton, B.A.: Detecting genus in vertex links for the fast enumeration of 3-manifold triangulations. In: Proceedings of ISSAC, pp. 59–66. ACM (2011)Google Scholar
  5. 5.
    Burton, B.A.: A new approach to crushing 3-manifold triangulations. Discrete Comput. Geom. 52(1), 116–139 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Burton, B.A., Budney, R., Pettersson, W., et al.: Regina: Software for 3-manifold topology and normal surface theory. https://regina-normal.github.io/ (1999–2019). Accessed 25 Oct 2019
  7. 7.
    Burton, B.A., Maria, C., Spreer, J.: Algorithms and complexity for Turaev-Viro invariants. J. Appl. Comput. Topol. 2(1-2), 33–53 (2018)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Burton, B.A., Ozlen, M.: A fast branching algorithm for unknot recognition with experimental polynomial-time behaviour. arXiv:1211.1079 (2012)
  9. 9.
    Edelsbrunner, H., Harer, J.L.: Computational topology: An introduction. American Mathematical Society, Providence, RI (2010)zbMATHGoogle Scholar
  10. 10.
    Freedman, M.H.: Complexity classes as mathematical axioms. Ann. of Math 170(2), 995–1002 (2009)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Frohman, C., Kania-Bartoszynska, J.: The quantum content of the normal surfaces in a three-manifold. J. Knot Theory Ramifications 17(8), 1005–1033 (2008)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)zbMATHGoogle Scholar
  13. 13.
    Hausmann, J.C.: Mod two homology and cohomology. Universitext. Springer, Cham (2014)CrossRefGoogle Scholar
  14. 14.
    Hodgson, C.D., Weeks, J.R.: Symmetries, isometries and length spectra of closed hyperbolic three-manifolds. Experiment. Math. 3(4), 261–274 (1994)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Huszár, K., Spreer, J.: 3-Manifold Triangulations with Small Treewidth. In: Proceedings of SOCG, pp. 44:1–44:20. LIPIcs (2019)Google Scholar
  16. 16.
    Huszár, K., Spreer, J., Wagner, U.: On the treewidth of triangulated 3-manifolds. In: Proceddings of SOCG, pp. 46:1–46:15. LIPIcs (2018)Google Scholar
  17. 17.
    Jaco, W., Rubinstein, J.H.: 0-Efficient Triangulations of 3-Manifolds. J. Differential Geom. 65(1), 61–168 (2003)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Kauffman, L.H., Lins, S.: Computing Turaev-Viro invariants for \(3\)-manifolds. Manuscripta Math. 72(1), 81–94 (1991)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Kirby, R., Melvin, P.: Local surgery formulas for quantum invariants and the Arf invariant. Geom. Topol. Monogr. pp. (7):213–233 (2004)Google Scholar
  20. 20.
    Kuperberg, G.: Algorithmic homeomorphism of \(3\)-manifolds as a corollary of geometrization. Pacific J. Math. 301(1), 189–241 (2019)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Lidl, R., Niederreiter, H.: Finite Fields. Cambridge University Press, Cambridge (1997)zbMATHGoogle Scholar
  22. 22.
    Maria, C., Purcell, J.S.: Treewidth, crushing, and hyperbolic volume. Algebraic Geom. Topology 19(5), 2625–2652.  https://doi.org/10.2140/agt.2019.19.2625 (2019) MathSciNetCrossRefGoogle Scholar
  23. 23.
    Maria, C., Spreer, J.: Admissible colourings of 3-manifold triangulations for Turaev-Viro type invariants. In: Proceedings of ESA, pp. 64:1–64:16 (2016)Google Scholar
  24. 24.
    Maria, C., Spreer, J.: Classification of normal curves on a tetrahedron. In: SOCG:YRF – Collection of Abstracts (2016)Google Scholar
  25. 25.
    Maria, C., Spreer, J.: A polynomial time algorithm to compute quantum invariants of 3-manifolds with bounded first betti number. In: Proceedings of SODA, pp. 2721–2732 (2017)Google Scholar
  26. 26.
    Markov, A.: The insolubility of the problem of homeomorphy. Dokl. Akad. Nauk SSSR 121, 218–220 (1958)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Matveev, S.: Transformations of special spines, and the Zeeman conjecture. Izv. Akad. Nauk SSSR Ser. Mat. 51(5), 1104–1116, 1119 (1987)Google Scholar
  28. 28.
    Matveev, S.: Algorithmic Topology and Classification of 3-Manifolds. Springer, Berlin (2003)CrossRefGoogle Scholar
  29. 29.
    Matveev, S., et al.: Manifold recognizer. http://www.matlas.math.csu.ru/?page=recognizer (2012). Accessed 25 Oct 2019
  30. 30.
    Mohar, B.: A linear time algorithm for embedding graphs in an arbitrary surface. SIAM J. Discrete Math. 12(1), 6–26 (1999)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Morgan, J., Tian, G.: The geometrization conjecture. American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA (2014)Google Scholar
  32. 32.
    Norine, S., Seymour, P.D., Thomas, R., Wollan, P.: Proper minor-closed families are small. J. Comb 96(5), 754–757 (2006)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Piergallini, R.: Standard moves for standard polyhedra and spines. Rend. Circ. Mat. Palermo (2) Suppl. (18), 391–414 (1988)Google Scholar
  34. 34.
    Schleimer, S., de Mesmay, A., Purcell, J., Sedgwick, E.: On the tree-width of knot diagrams. Journal of Computational Geometry 10(1), 164–180 (2019)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Sokolov, M.: Which lens spaces are distinguished by Turaev–Viro invariants? Mathematical Notes 61(3), 468–470 (1997)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Stillwell, J.: Classical topology and combinatorial group theory, Graduate Texts in Mathematics, vol. 72, Springer-Verlag, second edn. Springer-Verlag, New York (1993)CrossRefGoogle Scholar
  37. 37.
    Storjohann, A.: Algorithms for matrix canonical forms. Ph.D. thesis, ETH Zürich (2000)Google Scholar
  38. 38.
    Turaev, V.G.: Quantum invariants of knots and 3-manifolds, de Gruyter Studies in Mathematics, vol. 18, revised edn. Walter de Gruyter & Co., Berlin (2010)CrossRefGoogle Scholar
  39. 39.
    Turaev, V.G., Viro, O.Y.: State sum invariants of \(3\)-manifolds and quantum \(6j\)-symbols. Topology 31(4), 865–902 (1992)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Yamada, S.: The absolute value of the Chern-Simons-Witten invariants of lens spaces. J. Knot Theory Ramifications 4(2), 319–327 (1995)MathSciNetCrossRefGoogle Scholar

Copyright information

© SFoCM 2019

Authors and Affiliations

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

Personalised recommendations