Osaka Journal of Mathematics

Knots with Hopf crossing number at most one

Maciej Mroczkowski

Full-text: Open access

Abstract

We consider diagrams of links in $S^2$ obtained by projection from $S^3$ with the Hopf map and the minimal crossing number for such diagrams. Knots admitting diagrams with at most one crossing are classified. Some properties of these knots are exhibited. In particular, we establish which of these knots are algebraic and, for such knots, give an answer to a problem posed by Fiedler in [3].

Article information

Source
Osaka J. Math., Volume 57, Number 2 (2020), 279-304.

Dates
First available in Project Euclid: 6 April 2020

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1586160079

Mathematical Reviews number (MathSciNet)
MR4081733

Zentralblatt MATH identifier
07196679

Subjects
Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57M27: Invariants of knots and 3-manifolds

Citation

Mroczkowski, Maciej. Knots with Hopf crossing number at most one. Osaka J. Math. 57 (2020), no. 2, 279--304. https://projecteuclid.org/euclid.ojm/1586160079


Export citation

References

  • J.C. Cha and C. Livingston: KnotInfo: Table of Knot Invariants, available at http://www.indiana.edu/~knotinfo, May 16, 2018.
  • D. Eisenbud and W. Neumann: Three-dimensional Link Theory and Invariants of Plane Curve Singularities, Ann. Math. Studies 110, Princeton University Press, Princeton, NJ, 1985.
  • T. Fiedler: Algebraic links and the Hopf Fibration, Topology 30 (1991), 259–265.
  • V.F. R. Jones: Hecke Algebra Representations of Braid Groups and Link Polynomials, Ann. of Math. (2) 126, (1987), 335–388.
  • L.H. Kauffman: State models and the Jones polynomial, Topology 26 (1987), 395–407.
  • H. Morton: The coloured Jones function and Alexander polynomial for torus knots, Math. Proc. Cambridge Philos. Soc. 117 (1995), 129–135.
  • M. Mroczkowski and M. Dabkowski: KBSM of the product of a disk with two holes and $S^1$, Topology App. 156 (2009), 1831–1849.
  • D. Rolfsen: Knots and Links, Publish or Perish, Berkeley Calif., 1976.
  • R. Scharein: KnotPlot, available at http://www.knotplot.com, May 16 2018.
  • A.T. Tran: The strong AJ conjecture for cables of torus knots, J. Knot Theory Ramifications 24, (2015), 1550072, 11pp.
  • Le Dung Trang: Sur les noeuds algebriques, Compositio Math. 25 (1972), 281–321.
  • V.G. Turaev: Shadow links and face models of statistical mechanics, J. Differential Geom. 36 (1992), 35–74.