# Braidoids

## TL;DR

This paper introduces braidoids, a new mathematical structure generalizing classical braids, and establishes foundational theorems including an Alexander-type theorem and a Markov-type theorem for braidoids.

## Contribution

It defines braidoids, develops their basic properties, and proves key theorems linking braidoids to knotoids, expanding the mathematical framework of braid theory.

## Key findings

- Introduced the concept of braidoids.
- Developed a closure operation for braidoids.
- Proved an Alexander-type theorem for braidoids.

## Abstract

Braidoids generalize the classical braids and form a counterpart theory to the theory of planar knotoids, just as the theory of braids does for the theory of knots. In this paper, we introduce basic notions of braidoids, a closure operation for braidoids, we prove an analogue of the Alexander theorem, that is, an algorithm that turns a knotoid into a braidoid, and we formulate and prove a geometric analogue of the Markov theorem for braidoids using the $L$-moves.