Klein links and related torus links

Alvarado, Enrique; Beres, Steven; Coufal, Vesta; Hlavacek, Kaia; Pereira, Joel; Reeves, Brandon

doi:10.2140/involve.2016.9.347

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Klein links and related torus links

Enrique Alvarado, Steven Beres, Vesta Coufal, Kaia Hlavacek, Joel Pereira and Brandon Reeves

Vol. 9 (2016), No. 2, 347–359

DOI: 10.2140/involve.2016.9.347

Abstract

In this paper, we present our constructions and results leading up to our discovery of a class of Klein links that are not equivalent to any torus links. In particular, we calculate the number and types of components in a $K_{p, q}$ Klein link and show that $K_{p, p} \equiv K_{p, p - 1}$ , $K_{p, 2} \equiv T_{p - 1, 2}$ , and $K_{2 p, 2 p} \equiv T_{2 p, p}$ . Finally, we show that in contrast to the fact that every Klein knot is a torus knot, no Klein link $K_{p, p}$ , where $p \geq 5$ is odd, is equivalent to a torus link.

Keywords

knot theory, Klein links, torus links

Mathematical Subject Classification 2010

Primary: 57M25

Milestones

Received: 3 January 2015

Revised: 23 February 2015

Accepted: 26 February 2015

Published: 2 March 2016

Communicated by Colin Adams

Authors

Enrique Alvarado
Department of Mathematics Washington State University Pullman, WA 99164 United States

Steven Beres
Department of Mathematics Gonzaga University Spokane, WA 99258 United States

Vesta Coufal
Department of Mathematics Gonzaga University Spokane, WA 99258 United States

Kaia Hlavacek
Department of Mathematics Gonzaga University Spokane, WA 99258 United States

Joel Pereira
Department of Mathematics Gonzaga University Spokane, WA 99258 United States

Brandon Reeves
Department of Economics University of Wisconsin Madison, WI 53706 United States

Vol. 9, No. 2, 2016