Document Open Access Logo

Admissible Colourings of 3-Manifold Triangulations for Turaev-Viro Type Invariants

Authors Clément Maria, Jonathan Spreer

PDF
Thumbnail PDF

File

LIPIcs.ESA.2016.64.pdf
  • Filesize: 1.32 MB
  • 16 pages

Document Identifiers

Author Details

Clément Maria
Jonathan Spreer

Cite As Get BibTex

Clément Maria and Jonathan Spreer. Admissible Colourings of 3-Manifold Triangulations for Turaev-Viro Type Invariants. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 64:1-64:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016) https://doi.org/10.4230/LIPIcs.ESA.2016.64

Abstract

Turaev-Viro invariants are amongst the most powerful tools to distinguish 3-manifolds. They are invaluable for mathematical software, but current algorithms to compute them rely on the enumeration of an extremely large set of combinatorial data defined on the triangulation, regardless of the underlying topology of the manifold.

In the article, we propose a finer study of these combinatorial data, called admissible colourings, in relation with the cohomology of the manifold. We prove that the set of admissible colourings to be considered is substantially smaller than previously known, by furnishing new upper bounds on its size that are aware of the topology of the manifold. Moreover, we deduce new topology-sensitive enumeration algorithms based on these bounds. 

The paper provides a theoretical analysis, as well as a detailed experimental study of the approach. We give strong experimental evidence on large manifold censuses that our upper bounds are tighter than the previously known ones, and that our algorithms outperform significantly state of the art implementations to compute Turaev-Viro invariants.

Subject Classification

Keywords
  • low-dimensional topology
  • triangulations of 3-manifolds
  • cohomology theory
  • Turaev-Viro invariants
  • combinatorial algorithms

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. Benjamin A. Burton. Structures of small closed non-orientable 3-manifold triangulations. J. Knot Theory Ramifications, 16(5):545-574, 2007. Google Scholar
  2. Benjamin A. Burton. Detecting genus in vertex links for the fast enumeration of 3-manifold triangulations. In Proceedings of ISSAC, pages 59-66. ACM, 2011. Google Scholar
  3. Benjamin A. Burton. A new approach to crushing 3-manifold triangulations. Discrete Comput. Geom., 52(1):116-139, 2014. Google Scholar
  4. Benjamin A. Burton, Ryan Budney, Will Pettersson, et al. Regina: Software for 3-manifold topology and normal surface theory. http://regina.sourceforge.net/, 1999-2014. Google Scholar
  5. Benjamin A. Burton, Clément Maria, and Jonathan Spreer. Algorithms and complexity for Turaev-Viro invariants. In Proceedings of ICALP 2015, pages 281-293. Springer, 2015. Google Scholar
  6. Benjamin A. Burton and Melih Ozlen. A fast branching algorithm for unknot recognition with experimental polynomial-time behaviour. arXiv:1211.1079, 2012. Google Scholar
  7. Allen Hatcher. Algebraic Topology. Cambridge University Press, Cambridge, 2002. http://www.math.cornell.edu/~hatcher/AT/ATpage.html. Google Scholar
  8. Craig D. Hodgson and Jeffrey R. Weeks. Symmetries, isometries and length spectra of closed hyperbolic three-manifolds. Experiment. Math., 3(4):261-274, 1994. Google Scholar
  9. William Jaco and J. Hyam Rubinstein. 0-efficient triangulations of 3-manifolds. J. Differential Geom., 65(1):61-168, 2003. Google Scholar
  10. Robion Kirby and Paul Melvin. The 3-manifold invariants of Witten and Reshetikhin-Turaev for sl(2,C). Invent. Math., 105(3):473-545, 1991. URL: http://dx.doi.org/10.1007/BF01232277.
  11. Robion Kirby and Paul Melvin. Local surgery formulas for quantum invariants and the Arf invariant. Geom. Topol. Monogr., pages (7):213-233, 2004. URL: http://dx.doi.org/10.2140/gtm.2004.7.213.
  12. Sergei Matveev. Algorithmic Topology and Classification of 3-Manifolds. Number 9 in Algorithms and Computation in Mathematics. Springer, Berlin, 2003. Google Scholar
  13. Sergei Matveev et al. Manifold recognizer. http://www.matlas.math.csu.ru/?page=recognizer, accessed August 2012. Google Scholar
  14. Justin Roberts. Skein theory and Turaev-Viro invariants. Topology, 34(4):771-787, 1995. URL: http://dx.doi.org/10.1016/0040-9383(94)00053-0.
  15. Vladimir G. Turaev. Quantum Invariants of Knots and 3-Manifolds, volume 18 of de Gruyter Studies in Mathematics. Walter de Gruyter &Co., Berlin, revised edition, 2010. URL: http://dx.doi.org/10.1515/9783110221848.
  16. Vladimir G. Turaev and Oleg Y. Viro. State sum invariants of 3-manifolds and quantum 6j-symbols. Topology, 31(4):865-902, 1992. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact