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A New Way to Represent Links. One-Dimensional Formalism and Untangling Technology

A New Way to Represent Links. One-Dimensional Formalism and Untangling Technology An alternative link representation different from planar diagrams is discussed. Isotopy classes of unordered nonoriented links are realized as central elements of a monoid presented explicitly by a finite number of generators and relations. The group presented by two generators and three relations [[a,b],a 2 ba −2]=[[a,b],b 2 ab −2]=[[a,b],[a −1,b −1]]=1, where [x,y]=xyx −1 y −1, is proved to have a commutator subgroup isomorphic to the braid group on infinitely many strands. A new partial algorithm for unknot recognition is constructed. Experiments show that the algorithm allows the untangling of unknots whose planar diagram has hundreds of crossings. Here 'untangling' means 'finding an isotopy to the circle'. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Applicandae Mathematicae Springer Journals

A New Way to Represent Links. One-Dimensional Formalism and Untangling Technology

Acta Applicandae Mathematicae , Volume 69 (3) – Oct 19, 2004

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References (10)

Publisher
Springer Journals
Copyright
Copyright © 2001 by Kluwer Academic Publishers
Subject
Mathematics; Mathematics, general; Computer Science, general; Theoretical, Mathematical and Computational Physics; Complex Systems; Classical Mechanics
ISSN
0167-8019
eISSN
1572-9036
DOI
10.1023/A:1014299416618
Publisher site
See Article on Publisher Site

Abstract

An alternative link representation different from planar diagrams is discussed. Isotopy classes of unordered nonoriented links are realized as central elements of a monoid presented explicitly by a finite number of generators and relations. The group presented by two generators and three relations [[a,b],a 2 ba −2]=[[a,b],b 2 ab −2]=[[a,b],[a −1,b −1]]=1, where [x,y]=xyx −1 y −1, is proved to have a commutator subgroup isomorphic to the braid group on infinitely many strands. A new partial algorithm for unknot recognition is constructed. Experiments show that the algorithm allows the untangling of unknots whose planar diagram has hundreds of crossings. Here 'untangling' means 'finding an isotopy to the circle'.

Journal

Acta Applicandae MathematicaeSpringer Journals

Published: Oct 19, 2004

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