Over then Under Tangles

Dror Bar-Natan; Zsuzsanna Dancso; Roland van der Veen

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arXiv · 2007.09828

Over then Under Tangles

Dror Bar-Natan·Zsuzsanna Dancso·Roland van der Veen

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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 any tangle diagram could be brought to OU form. Unfortunately, the algorithm is flawed. However, by analyzing cases in which it does succeed 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" which appear 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,

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Dror Bar-Natan, Zsuzsanna Dancso, Roland van der Veen. 2020-07-20. Over then Under Tangles. https://arxiv.org/abs/2007.09828

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