Martin David Palmer -- Publication -- Triple-crossing number and moves on triple-crossing link diagrams
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Triple-crossing number and moves on triple-crossing link diagrams
with Colin Adams and Jim Hoste

Journal of Knot Theory and Its Ramifications vol. 28 no. 11 (2019) 1940001 (20 pp.)
arXiv: abspdf
v1

Abstract

Every link in the 3-sphere has a projection to the plane where the only singularities are pairwise transverse triple points. The associated diagram, with height information at each triple point, is a triple-crossing diagram of the link. We give a set of diagrammatic moves on triple-crossing diagrams analogous to the Reidemeister moves on ordinary diagrams. The existence of n-crossing diagrams for every n>1 allows the definition of the n-crossing number. We prove that for any nontrivial, nonsplit link, other than the Hopf link, its triple-crossing number is strictly greater than its quintuple-crossing number.

Click here for the notes of a talk that I gave, based on the results of this paper, on 7 September 2020 at (virtually) the Knots and representation theory seminar, Moscow.

Our (five) Reidemeister moves for 3-crossing diagrams are drawn on page 8 of these notes, and pages 9–12 contain a sketch proof of our main result — that they form a complete set of Reidemeister moves for 3-crossing diagrams.