Passing from plat closure to standard closure of braids in R^3, in handlebodies and in thickened surfaces
Year: 2024
Type: article
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 ℝ3, in handlebodies and in thickened surfaces, and we give for each one a detailed construction of the relative algorithm, with proofs of the results and the calculation of the computational complexity. Indeed, we show that the algorithm has quadratic computational complexity in the number of crossings and loop generators of the braid when passing from plat to standard closure, while it has linear computational complexity in the number of crossings generators of the braid when passing from standard to plat closure. The article includes numerous illustrations to facilitate the understanding of the results, as well as an example of implementation of the algorithm for the general case.
Source: Ars Mathematica Contemporanea
Authors: Paolo Cavicchioli, Sofia Lambropoulou
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W4405644161