A braidoid equivalence for spherical knotoids

TL;DR

This paper introduces a refined equivalence relation for braidoid diagrams, establishing a correspondence between spherical knotoids and their braidoid representations, thus advancing the understanding of knotoid equivalences.

Contribution

It refines the concept of L-equivalence for braidoid diagrams to establish an equivalence theorem for spherical knotoids.

Findings

Refined L-equivalence for braidoids.

Established an equivalence theorem for spherical knotoids.

Connected braidoid diagrams with knotoid classifications.

Abstract

Braidoids form a counterpart theory to the theory of planar knotoids, just as braids do for three-dimensional links. As such, planar knotoid diagrams represent the same knotoid in $\mathbb{R}^2$ if and only if they can be presented as the closure of two labeled braidoid diagrams related by an equivalence relation, named $L$-equivalence. In this paper, we refine the notion of $L$-equivalence of braidoid diagrams in order to obtain an equivalence theorem for (multi)-knotoid diagrams in $S^2$ when represented as the closure of labeled braidoid diagrams.