Knots with Hopf crossing number at most one. (English) Zbl 1437.57011

Summary: 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 T. Fiedler in [Topology 30, No. 2, 259–265 (1991; Zbl 0725.57002)].


57K10 Knot theory


Zbl 0725.57002


KnotPlot; KnotInfo
Full Text: arXiv Euclid


[1] J.C. Cha and C. Livingston: KnotInfo: Table of Knot Invariants, available at http://www.indiana.edu/ knotinfo, May 16, 2018.
[2] 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. · Zbl 0628.57002
[3] T. Fiedler: Algebraic links and the Hopf Fibration, Topology 30 (1991), 259-265. · Zbl 0725.57002
[4] V.F. R. Jones: Hecke Algebra Representations of Braid Groups and Link Polynomials, Ann. of Math. (2) 126, (1987), 335-388. · Zbl 0631.57005
[5] L.H. Kauffman: State models and the Jones polynomial, Topology 26 (1987), 395-407. · Zbl 0622.57004
[6] H. Morton: The coloured Jones function and Alexander polynomial for torus knots, Math. Proc. Cambridge Philos. Soc. 117 (1995), 129-135. · Zbl 0852.57007
[7] M. Mroczkowski and M. Dabkowski: KBSM of the product of a disk with two holes and \(S^1\), Topology App. 156 (2009), 1831-1849. · Zbl 1168.57010
[8] D. Rolfsen: Knots and Links, Publish or Perish, Berkeley Calif., 1976.
[9] R. Scharein: KnotPlot, available at http://www.knotplot.com, May 16 2018.
[10] A.T. Tran: The strong AJ conjecture for cables of torus knots, J. Knot Theory Ramifications 24, (2015), 1550072, 11pp. · Zbl 1339.57021
[11] Le Dung Trang: Sur les noeuds algebriques, Compositio Math. 25 (1972), 281-321. · Zbl 0245.14003
[12] V.G. Turaev: Shadow links and face models of statistical mechanics, J. Differential Geom. 36 (1992), 35-74. · Zbl 0773.57012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.