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Gauss-Lintel, an Algorithm Suite for Exploring Chord Diagrams

Part of the Lecture Notes in Computer Science book series (LNAI,volume 12833)

Abstract

Gauss diagrams, or more generally chord diagrams are a well-established tool in the study of topology of knots and of planar curves. In this paper we present a system description of Gauss-lintel, our implementation in SWI-Prolog of a suite of algorithms for exploring chord diagrams. Gauss-lintel employs a datatype which we call “lintel”, which is a list representation of an odd-even matching for the set of integers [0,...,2n–1], for efficiently generating Gauss diagrams and testing their properties, including one important property called realizability. We report on extensive experiments in generation and enumeration of various classes of Gauss diagrams, as well as on experimental testing of several published descriptions of realizability.

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Fig. 1.

Notes

  1. 1.

    A dynamic fact is a Prolog fact (atomic formula) which can be added or removed during run time. Using dynamic facts goes outside of the pure Prolog paradigm, but it is very useful for keeping information about the global state.

  2. 2.

    TaitCurves.

References

  1. The On-Line Encyclopedia of Integer Sequences. http://oeis.org/A264759

  2. Biryukov, O.N.: Parity conditions for realizability of Gauss diagrams. J. Knot Theor. Ramifications 28(1), 1950015 (2019)

    Google Scholar 

  3. Cairns, G., Elton, D.M.: The planarity porblem II. J. Knot Theor. Ramification 5(2), 137–144 (1996)

    CrossRef  Google Scholar 

  4. Chmutov, M., Hulse, T., Lum, A., Rowell, P.: Plane and spherical curves: an investigation of their invariants. In: Summer Mathematics Research Institute, REU 2006 Proceedings, Oregon State University (2006)

    Google Scholar 

  5. de Fraysseix, H., Ossona de Mendez, P.: A short proof of a Gauss problem. In: DiBattista, G. (ed.) GD 1997. LNCS, vol. 1353, pp. 230–235. Springer, Heidelberg (1997). https://doi.org/10.1007/3-540-63938-1_65

    CrossRef  Google Scholar 

  6. Dehn, M.: Über Kombinatorische Topologie. Acta Math. 67, 123–168 (1936)

    MathSciNet  CrossRef  Google Scholar 

  7. Gauss, C.F.: Werke, vol. VII. Tuebner, Leipzig (1900)

    MATH  Google Scholar 

  8. Grinblat, A., Lopatkin, V.: On realizabilty of Gauss diagrams and constructions of meanders. J. Knot Theor. Ramifications 29(5), 2050031 (2020)

    Google Scholar 

  9. Kauffman, L.H.: Virtual knot theory. Eur. J. Comb. 20(7), 663–691 (1999)

    MathSciNet  CrossRef  Google Scholar 

  10. Khan, A., Lisitsa, A., Vernitski, A.: Experimental mathematics approach to Gauss diagrams realizability. arxiv:2103.02102 (2021)

  11. Khan, A., Lisitsa, A., Vernitski, A.: Gauss-lint algorithms suite for Gauss diagrams generation and analysis. Zenodo (2021). https://doi.org/10.5281/zenodo.4574590

  12. Lovász, L., Marx, M.L.: A forbidden substructure characterization of Gauss codes. Bull. Am. Math. Soc. 82(1), 121–122 (1976)

    Google Scholar 

  13. Lyndon, R.C.: On Burnside’s problem. Trans. Am. Math. Soc. 77, 202–215 (1954)

    MathSciNet  MATH  Google Scholar 

  14. Rosenstiehl, P.: Solution algébrique du problème de Gauss sur la permutation des points d’intersection d’une ou plusieurs courbes fermées du plan. C.R. Acad. Sci. 283(série A), 551–553 (1976)

    Google Scholar 

  15. Shtylla, B., Traldi, L., Zulli, L.: On the realization of double occurrence words. Discrete Math. 309(6), 1769–1773 (2009)

    MathSciNet  CrossRef  Google Scholar 

  16. Valette, G.: A classification of spherical curves based on Gauss diagrams. Arnold Math. J. 2(3), 383–405 (2016). https://doi.org/10.1007/s40598-016-0049-3

    MathSciNet  CrossRef  MATH  Google Scholar 

  17. Wielemaker, J., Schrijvers, T., Triska, M., Lager, T.: SWI-prolog. Theor. Pract. Logic Program. 12(1–2), 67–96 (2012)

    MathSciNet  CrossRef  Google Scholar 

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Acknowledgements

This work was supported by the Leverhulme Trust Research Project Grant RPG-2019-313.

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Correspondence to Alexei Lisitsa .

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Khan, A., Lisitsa, A., Vernitski, A. (2021). Gauss-Lintel, an Algorithm Suite for Exploring Chord Diagrams. In: Kamareddine, F., Sacerdoti Coen, C. (eds) Intelligent Computer Mathematics. CICM 2021. Lecture Notes in Computer Science(), vol 12833. Springer, Cham. https://doi.org/10.1007/978-3-030-81097-9_16

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  • DOI: https://doi.org/10.1007/978-3-030-81097-9_16

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