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Detecting Unknots via Equational Reasoning, I: Exploration

  • Andrew Fish
  • Alexei Lisitsa
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8543)

Abstract

We explore the application of automated reasoning techniques to unknot detection, a classical problem of computational topology. We adopt a two-pronged experimental approach, using a theorem prover to try to establish a positive result (i.e. that a knot is the unknot), whilst simultaneously using a model finder to try to establish a negative result (i.e. that the knot is not the unknot). The theorem proving approach utilises equational reasoning, whilst the model finder searches for a minimal size counter-model. We present and compare experimental data using the involutary quandle of the knot, as well as comparing with alternative approaches, highlighting instances of interest. Furthermore, we present theoretical connections of the minimal countermodels obtained with existing knot invariants, for all prime knots of up to 10 crossings: this may be useful for developing advanced search strategies.

Keywords

Fundamental Group Homomorphic Image Automate Reasoning Automate Theorem Prove Automate Deduction 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Andreeva, M., Dynnikov, I., Koval, S., Polthier, K., Taimanov, I.: Book Knot Simplifier, http://www.javaview.de/services/knots/doc/description.html (accessed March 14, 2014)
  2. 2.
    The CADE ATP System Competition, The World Championship for Automated Theorem Proving, http://www.cs.miami.edu/~tptp/CASC/ (accessed June 07, 2013)
  3. 3.
    Burton, B.A., Olzen, M.: A fast branching algorithm for unknot recognizion with experimental polynomial-time behaviour. arXiv:1211.1079 [math.GT]Google Scholar
  4. 4.
    Birkhoff, G.: On the structure of abstract algebras. Proc. Cambridge Philos. Soc. 31, 433–454 (1935)CrossRefzbMATHGoogle Scholar
  5. 5.
    Caferra, R., Leitsch, A., Peltier, N.: Automated Model Building. Applied Logic Series, vol. 31. Kluwer (2004)Google Scholar
  6. 6.
    Scott Carter, J.: A survey of quandle ideas. arXiv:1002.4429 [math.GT]Google Scholar
  7. 7.
    Dynnikov, I.A.: Recognition algorithms in knot theory. Uspekhi Mat. Nauk 58(6(354)), 45–92 (2003)Google Scholar
  8. 8.
    Dynnikov, A.: Three-page link presentation and an untangling algorithm. In: Proc. of the International Conference Low-Dimensional Topology and Combinatorial Group Theory, Chelyabinsk, Kiev, July 31-August 7, pp. 112–130 (1999, 2000)Google Scholar
  9. 9.
    Fenn, R., Rourke, C.: Racks and Links in Codimension two. J. Knot Theory Ramifications 01, 343 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Goubault-Larrecq, J., Mackie, I.: Proof Theory and Automated Deduction. Applied Logic Series, vol. 6. Kluwer (2001)Google Scholar
  11. 11.
    Haken, W.: Theorie der Normal achen. Acta Math. 105, 245–375 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hass, J., Lagarias, J.C., Pippenger, N.: The computational complexity of knot and link problems. J. Assoc. Comput. Mach. 46(2), 185–211 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hempel, J.: Residual finiteness for 3-manifolds. In: Combinatorial Group Theory and Topology (Alta, Utah, 1984). Ann. of Math. Stud., vol. 111, pp. 379–396. Princeton Univ. Press, Princeton (1987)Google Scholar
  14. 14.
    Joyce, D.: A Classifying Invariant of Knots, the Knot Quandle. Journal of Pure and Applied Algebra 23, 37–65 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Joyce, D.: Simple Quandles. Journal of Algebra 79, 307–318 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kuperberg, G.: Knottedness is in NP, modulo GRH, Preprint, arXiv:1112.0845 (November 2011)Google Scholar
  17. 17.
    Unknot detection by equational reasoning, http://www.csc.liv.ac.uk/~alexei/unknots/ (accessed April 14, 2014)
  18. 18.
    McCune, W.: Prover9 and Mace4, http://www.cs.unm.edu/~mccune/mace4/
  19. 19.
    Winker, S.N.: Quandles, Knot Invariants and the N-fold Branching Cover. PhD Thesis, University of Illinois at Chicago (1984)Google Scholar
  20. 20.
    Reidemeister, K.: Elementare Begründung der Knotentheorie. Abh. Math. Sem. Univ. Hamburg 5, 24–32 (1926)CrossRefzbMATHGoogle Scholar
  21. 21.
    Thurston, W.P.: Three-dimensional manifolds, Kleinian groups and hyperbolic geometry. Bull. Amer. Math. Soc. (N.S.) 6(3), 357–381 (1982)Google Scholar
  22. 22.
    Cha, J.C., Livingston, C.: KnotInfo: Table of knot invariants, http://www.indiana.edu/~knotinfo (accessed January 2014)
  23. 23.
    Wu, Z.S.: Computable Invariants for Quandles. Thesis, Bard College, New York (2012)Google Scholar
  24. 24.
    Wallace, S.D.: Homomorphic images of link quandles. MA thesis, Houston, Texas (2004)Google Scholar
  25. 25.
    Manoim, B.: Toward an Online Knowledgebase for Knots and Quandles. Technical report, http://asclab.org/asc/sites/default/files/docs/Online%20database%20knots%20%26%20quandles.pdf
  26. 26.
    Rolfsen, D.: Knots and Links. AMS Chelsea Publishing (2004)Google Scholar
  27. 27.

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Andrew Fish
    • 1
  • Alexei Lisitsa
    • 2
  1. 1.School of Computing, Engineering and MathematicsUniversity of BrightonUK
  2. 2.Department of Computer ScienceThe University of LiverpoolUK

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