Passing from plat closure to standard closure of braids in $\mathbb{R}^3$, in handlebodies and in thickened surfaces

TL;DR

This paper presents an algorithmic method to convert between plat and standard closures of braids across different 3D manifolds, with efficiency analysis for each conversion direction.

Contribution

It introduces a quadratic-time algorithm for plat to standard closure conversion and a linear-time algorithm for the reverse, applicable in \\mathbb{R}^3, handlebodies, and thickened surfaces.

Findings

Quadratic complexity for plat to standard conversion.

Linear complexity for standard to plat conversion.

Applicable in various 3D manifold contexts.

Abstract

Given a knot or link in the form of plat closure of a braid, we describe an algorithm to obtain a braid representing the same knot or link with the standard closure, and vice-versa. We analyze the three cases of knots and links: in \(\mathbb{R}^3\), in handlebodies and in thickened surfaces. We show that the algorithm is quadratic in the number of crossings and loop generators of the braid when passing from plat to standard closure, while it is linear when passing from standard to plat closure.