Knotoids, Braidoids and Applications

Abstract

This paper is an introduction to the theory of braidoids. Braidoids are geometric objects analogous to classical braids, forming a counterpart theory to the theory of knotoids. We introduce these objects and their topological equivalences, and we conclude with a potential application to the study of proteins.

A Survey on Knotoids, Braidoids and Their Applications

“…In this section we review the fundamental notions of braidoids introduced by the first and the last listed authors [18,19]. Braidoids are defined so as to form a braided counterpart theory to the theory of planar knotoids.…”

“…The proof of the Alexander theorem by the last listed author [27,28] utilizes the L-braiding moves. In [18,19] the first and the last listed authors proved the following analogue of the Alexander theorem for (multi)-knotoids by utilizing these moves.…”

“…These algorithms are both based on the L-braidoiding moves. In Figure 15 we illustrate the steps of the algorithm that is given in [19]. As for the algorithm in [23] turning any virtual knot/link diagram into a virtual braid diagram, we start by rotating each crossing containing one or two up-arcs by π 2 or π, respectively, so that we end up with a knotoid diagram whose up-arcs are all free of any crossings.…”

“…The algorithm terminates in finite steps and results in a labeled braidoid diagram in Figure 15. The algorithm in [19] which is based on [28], uses braidoiding moves for up-arcs in crossings and is more appropriate for establishing Markov-type theorems for braidoids (see Theorem 6). Yet, an added value of the first algorithm is the following consequence: Any (multi)-knotoid diagram is isotopic to the uniform closure of a braidoid diagram [13,19].…”

“…Further, with the introduction of Lmoves on braidoids, which were originally defined for classical braids by the last author [27,28], a geometric analogue of the classical Markov theorem [31] for braidoids is enabled. In [13,18,14] a set of combinatorial elementary blocks for braidoids is also introduced, which in [18] are proposed for an algebraic encoding of the entanglement of open protein chains in 3-dimensional space.…”