Over then Under Tangles

TL;DR

This paper investigates the concept of Over-then-Under tangles, introduces a flawed algorithm for transforming tangles into this form, and derives a braid classification result from successful cases, extending it to virtual braids.

Contribution

It analyzes the idea of gliding to achieve OU form, provides a classification result, and extends the approach to virtual braids with an implementation.

Findings

The gliding algorithm is flawed but useful in certain cases.

A braid classification result is obtained from successful gliding cases.

The approach is extended to virtual braids and related to existing literature.

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,