DOI QR코드

DOI QR Code

Quantization of the Crossing Number of a Knot Diagram

  • KAWAUCHI, AKIO (Osaka City University Advanced Mathematical Institute) ;
  • SHIMIZU, AYAKA (Department of Mathematics, Gunma National College of Technology)
  • Received : 2014.01.28
  • Accepted : 2014.07.14
  • Published : 2015.09.23

Abstract

We introduce the warping crossing polynomial of an oriented knot diagram by using the warping degrees of crossing points of the diagram. Given a closed transversely intersected plane curve, we consider oriented knot diagrams obtained from the plane curve as states to take the sum of the warping crossing polynomials for all the states for the plane curve. As an application, we show that every closed transversely intersected plane curve with even crossing points has two independent canonical orientations and every based closed transversely intersected plane curve with odd crossing points has two independent canonical orientations.

Keywords

Crossing number;Oriented knot diagram;Plane curve;Warping crossing polynomial;Warping degree;Warping polynomial

References

  1. S. Fujimura, On the ascending number of knots, thesis, Hiroshima University, 1988.
  2. T. S. Fung, Immersions in knot theory, a dissertation, Columbia University, 1996.
  3. A. Kawauchi, A survey of knot theory, Birkhauser, (1996).
  4. A. Kawauchi, Lectures on knot theory (in Japanese), Kyoritsu shuppan Co. Ltd, 2007.
  5. A. Kawauchi, On a complexity of a spatial graph, in: Knots and softmatter physics, Topology of polymers and related topics in physics, mathematics and biology, Bussei Kenkyu, 92-1(2009-4), 16-19.
  6. W. B. R. Lickorish and K. C. Millett, A polynomial invariant of oriented links, Topology, 26(1987), 107-141.
  7. M. Okuda, A determination of the ascending number of some knots, thesis, Hiroshima University, 1988.
  8. M. Ozawa, Ascending number of knots and links, J. Knot Theory Ramifications, 19(2010), 15-25. https://doi.org/10.1142/S0218216510007723
  9. M. Polyak, Minimal generating sets of Reidemeister moves, Quantum Topology, 1(2010), 399-411.
  10. A. Shimizu, The warping degree of a knot diagram, J. Knot Theory Ramifications, 19(2010), 849-857. https://doi.org/10.1142/S0218216510008194
  11. A. Shimizu, The warping degree of a link diagram, Osaka J. Math., 48(2011), 209-231.
  12. A. Shimizu, The warping polynomial of a knot diagram, J. Knot Theory Ramifications, 21(2012), 1250124. https://doi.org/10.1142/S0218216512501246

Cited by

  1. The rank of a warping matrix vol.206, 2016, https://doi.org/10.1016/j.topol.2016.04.003
  2. On the orientations of monotone knot diagrams vol.26, pp.10, 2017, https://doi.org/10.1142/S0218216517500535