Martin David Palmer -- Paper -- 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

To appear in Journal of Knot Theory and Its Ramifications
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.