Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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Colourings and the Alexander Polynomial

  • Camacho, Luis (Departamento de Matematica, Faculdade de Ciencias Exatas e da Engenharia, Universidade da Madeira) ;
  • Dionisio, Francisco Miguel (SQIG - Instituto de Telecomunicacoes and Department of Mathematics, Instituto Superior Tecnico, Universidade de Lisboa) ;
  • Picken, Roger (Center for Mathematical Analysis, Geometry and Dynamical Systems and Department of Mathematics, Instituto Superior Tecnico, Universidade de Lisboa)
  • Received : 2015.01.13
  • Accepted : 2015.06.17
  • Published : 2016.09.23
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Abstract

Using a combination of calculational and theoretical approaches, we establish results that relate two knot invariants, the Alexander polynomial, and the number of quandle colourings using any finite linear Alexander quandle. Given such a quandle, specified by two coprime integers n and m, the number of colourings of a knot diagram is given by counting the solutions of a matrix equation of the form AX = 0 mod n, where A is the m-dependent colouring matrix. We devised an algorithm to reduce A to echelon form, and applied this to the colouring matrices for all prime knots with up to 10 crossings, finding just three distinct reduced types. For two of these types, both upper triangular, we found general formulae for the number of colourings. This enables us to prove that in some cases the number of such quandle colourings cannot distinguish knots with the same Alexander polynomial, whilst in other cases knots with the same Alexander polynomial can be distinguished by colourings with a specific quandle. When two knots have different Alexander polynomials, and their reduced colouring matrices are upper triangular, we find a specific quandle for which we prove that it distinguishes them by colourings.

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References

  1. J. W. Alexander, Topological invariants of knots and links, Trans. Amer. Math. Soc., 30(1928), 275-306. https://doi.org/10.1090/S0002-9947-1928-1501429-1
  2. Y. Bae, Coloring link diagrams by Alexander quandles, J. Knot Theory Ramifications, 21(10)(2012), 1250094, 13 pp. https://doi.org/10.1142/S0218216512500940
  3. A. T. Butson and B. M. Stewart, Systems of linear congruences, Can. J. Math., 7(1955), 358-368. https://doi.org/10.4153/CJM-1955-039-2
  4. J. Carter, A survey of quandle ideas, Kauffman, Louis H. (ed.) et al., Introductory lectures on knot theory. Series on Knots and Everything 46(2012), 22-53.
  5. J. S. Carter, D. Jelsovsky, S. Kamada, L. Langford and M. Saito, Quandle cohomology and state-sum invariants of knotted curves and surfaces, Trans. Amer. Math. Soc., 355(1999), 3947-3989.
  6. J. S. Carter, D. Jelsovsky, S. Kamada and M. Saito, Computations of quandle cocycle invariants of knotted curves and surfaces, Adv. Math., 157(1)(2001), 36-94. https://doi.org/10.1006/aima.2000.1939
  7. R. Crowell and R. Fox, Introduction to knot theory, Graduate Texts in Mathematics. Springer-Verlag (1977).
  8. F. M. Dionisio and P. Lopes, Quandles at finite temperatures II, J. Knot Theory Ramifications, 12(8)(2003), 1041-1092. https://doi.org/10.1142/S0218216503002949
  9. N. Gilbert and T. Porter, Knots and surfaces, Oxford Science Publications. Oxford University Press (1994).
  10. C. Hayashi, M. Hayashi and K. Oshiro, On linear n-colorings for knots, J. Knot Theory Ramifications 21(14)(2012), 1250123, 13 pp. https://doi.org/10.1142/S0218216512501234
  11. A. Inoue, Quandle homomorphisms of knot quandles to Alexander quandles, J. Knot Theory Ramifications, 10(6)(2001), 813-821. https://doi.org/10.1142/S0218216501001177
  12. D. Joyce, A classifying invariant of knots, the knot quandle, J. Pure Appl. Algebra, 23(1)(1982), 37-65. https://doi.org/10.1016/0022-4049(82)90077-9
  13. L. H. Kauffman Knots and Physics, K & E Series on Knots and Everything. World Scientific (1991).
  14. L. H. Kauffman, Virtual knot theory, Eur. J. Comb., 20(7)(1996), 663-691. https://doi.org/10.1006/eujc.1999.0314
  15. A. Kawauchi, A Survey of Knot Theory, Birkhauser Basel (1996).
  16. C. Livingston, Knot Theory, The Carus Mathematical Monographs Number 24, Mathematical Association of America (1993).
  17. P. Lopes, Quandles at nite temperatures I, J. Knot Theory Ramifications, 12(2)(2003), 159-186. https://doi.org/10.1142/S0218216503002378
  18. S. V. Matveev, Distributive groupoids in knot theory, Mat. Sb. (N.S.), 119(161)(1)(1982), 78-88, 160.
  19. S. Nelson, Classi cation of nite Alexander quandles, Topology Proc., 27(1)(2003), 245-258.
  20. M. Newman, Integral matrices, Pure and Applied Mathematics. Elsevier Science (1972).
  21. D. Rolfsen, Knots and Links, AMS Chelsea Publishing. American Mathematical Society (2003).