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Over then under tangles

Dror Bar-Natan

Department of Mathematics, University of Toronto, Toronto Ontario, M5S 2E4, Canada

E-mail Address: drorbn@math.toronto.edu

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Zsuzsanna Dancso

School of Mathematics and Statistics, The University of Sydney, Eastern Ave, Camperdown NSW 2006, Australia

E-mail Address: zsuzsanna.dancso@sydney.edu.au

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, and

Roland van der Veen

University of Groningen, Bernoulli Institute, P.O. Box 407, 9700 AK Groningen, The Netherlands

E-mail Address: roland.mathematics@gmail.com

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https://doi.org/10.1142/S0218216523400035Cited by:1 (Source: Crossref)

This article is part of the issue:

Special Issue: Dedicated to the work and memory of Vaughan F. R. Jones (Part 1)
Guest Editors: S. Lambropoulou and L. H. Kauffman

Abstract

Over-then-Under (OU) tangles are oriented tangles whose strands travel through all of their over crossings before any under crossings. In this paper, we discuss the idea of gliding: an algorithm by which tangle diagrams could be brought to OU form. By analyzing cases in which the algorithm converges, we obtain a braid classification result, which we also extend to virtual braids, and provide a Mathematica implementation. We discuss other instances of successful “gliding ideas” in the literature — sometimes in disguise — such as the Drinfel’d double construction, Enriquez’s work on quantization of Lie bialgebras, and Audoux and Meilhan’s classification of welded homotopy links.

Keywords:

AMSC: 57K12, 20F36, 20F38

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