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Triple-crossing number and moves on triple-crossing link diagrams

Colin Adams

Williams College, Williamstown, MA 01267, USA

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Jim Hoste

Pitzer College, Claremont, CA 91711, USA

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, and

Martin Palmer

Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany

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https://doi.org/10.1142/S0218216519400017Cited by:8 (Source: Crossref)

This article is part of the issue:

Special Issue: Knots, Low-Dimensional Topology and Applications — Part I

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 $n$ greater than one 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.

Keywords:

AMSC: 57M25

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