Vol. 8, No. 3, 2015

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ISSN: 1944-4184 (e-only)
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A new partial ordering of knots

Arazelle Mendoza, Tara Sargent, John Travis Shrontz and Paul Drube

Vol. 8 (2015), No. 3, 447–466
Abstract

Our research concerns how knots behave under crossing changes. In particular, we investigate a partial ordering of alternating knots that results from performing crossing changes. A similar ordering was originally introduced by Kouki Taniyama in the paper “A partial order of knots”. We amend Taniyama’s partial ordering and present theorems about the structure of our ordering for more complicated knots. Our approach is largely graph theoretic, as we translate each knot diagram into one of two planar graphs by checkerboard coloring the plane. Of particular interest are the class of knots known as pretzel knots, as well as knots that have only one direct minor in the partial ordering.

Keywords
knots, links, crossing changes
Mathematical Subject Classification 2010
Primary: 57M25
Secondary: 57M27
Milestones
Received: 21 June 2013
Revised: 30 March 2014
Accepted: 2 April 2014
Published: 5 June 2015

Communicated by Józef H. Przytycki
Authors
Arazelle Mendoza
Christopher Newport University
Newport News, VA 23606
United States
Tara Sargent
Clarke University
Dubuque, IA 52001
United States
John Travis Shrontz
University of Alabama in Huntsville
Huntsville, AL 35816
United States
Paul Drube
Department of Mathematics and Computer Science
Valparaiso University
1900 Chapel Drive
Valparaiso, IN 46383
United States