Abstract
This is a report on our ongoing research on a combinatorial approach to knot recognition, using coloring of knots by certain algebraic objects called quandles. The aim of the paper is to summarize the mathematical theory of knot coloring in a compact, accessible manner, and to show how to use it for computational purposes. In particular, we address how to determine colorability of a knot, and propose to use SAT solving to search for colorings. The computational complexity of the problem, both in theory and in our implementation, is discussed. In the last part, we explain how coloring can be utilized in knot recognition.
D. Stanovský—Partially supported by the GAČR grant 13-01832S.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Adams, C.: The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. Amer. Math. Soc., Providence (1994)
Andruskiewitsch, N., Graña, M.: From racks to pointed Hopf algebras. Adv. Math. 178(2), 177–243 (2003)
Bae, Y.: Coloring link diagrams by Alexander Quandles. J. Knot Theory Ramifications 21(10), 1250094 (2012)
Buck, D., Flapan, E. (eds.): Applications of Knot Theory. Proceedings of Symposia in Applied Mathematics. Amer. Math. Soc., Providence (2009)
Cha, J.C., Livingston, C.: KnotInfo: Table of knot invariants. http://www.indiana.edu/~knotinfo
Clark, W.E., Elhamdadi, M., Saito, M., Yeatman, T.: Quandle colorings of knots and applications. J. Knot Theory Ramifications 23(6), 1450035 (2014)
Clark, W.E., Saito, M., Vendramin, L.: Quandle coloring and cocycle invariants of composite knots and abelian extensions. http://arxiv.org/abs/1407.5803
Fish, A., Lisitsa, A.: Detecting unknots via equational reasoning, I: Exploration. In: Watt, S.M., Davenport, J.H., Sexton, A.P., Sojka, P., Urban, J. (eds.) CICM 2014. LNCS, vol. 8543, pp. 76–91. Springer, Heidelberg (2014)
Hass, J., Lagarias, J., Pippenger, N.: The computational complexity of knot and link problems. J. ACM 46, 185–211 (1999)
Henrich, A., Kauffman, L.: Unknotting Unknots. Am. Math. Monthly 121(5), 379–390 (2014)
Hulpke, A., Stanovský, D., Vojtěchovský, P.: Connected quandles and transitive groups. To appear in J. Pure Appl. Algebra. http://arxiv.org/abs/1409.2249
Joyce, D.: Classifying invariant of knots, the knot quandle. J. Pure Appl. Algebra 23, 37–65 (1982)
Joyce, D.: Simple quandles. J. Algebra 79, 307–318 (1982)
Kuperberg, G.: Knottedness is in NP, modulo GRH. Adv. Math. 256, 493–506 (2014)
Lackenby, M.: A polynomial upper bound on Reidemeister moves. To appear in Annals Math. http://arxiv.org/abs/1302.0180
Larose, B., Zádori, L.: Taylor terms, constraint satisfaction and the complexity of polynomial equations over finite algebras. Internat. J. Algebra Comput. 16(3), 563–581 (2006)
Matveev, S.V.: Distributive groupoids in knot theory. Math. USSR - Sbornik 47(1), 73–83 (1984)
Nelson, S.: The combinatorial revolution in knot theory. Notices Amer. Math. Soc. 58(11), 1553–1561 (2011)
Ohtsuki, T.: Problems on Invariants of Knots and 3-manifolds. Geom. Topol. Monogr., vol. 4. Geom. Topol. Publ., Coventry (2002)
Silver, D.S.: Knot theory’s odd origins. Am. Sci. 94, 158–165 (2006)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Fish, A., Lisitsa, A., Stanovský, D. (2015). A Combinatorial Approach to Knot Recognition. In: Horne, R. (eds) Embracing Global Computing in Emerging Economies. EGC 2015. Communications in Computer and Information Science, vol 514. Springer, Cham. https://doi.org/10.1007/978-3-319-25043-4_7
Download citation
DOI: https://doi.org/10.1007/978-3-319-25043-4_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-25042-7
Online ISBN: 978-3-319-25043-4
eBook Packages: Computer ScienceComputer Science (R0)