A Survey on Knotoids, Braidoids and Their Applications

Abstract

This paper is a survey on the theory of knotoids and braidoids. Knotoids are open ended knot diagrams in surfaces and braidoids are geometric objects analogous to classical braids, forming a counterpart theory to the theory of knotoids in the plane. We survey through the fundamental notions and existing works on these objects as well as their applications in the study of proteins. IntroductionThe theory of knotoids was introduced by Turaev [33] in 2010. A knotoid diagram is an oriented curve with two endpoints, immersed in an oriented surface, and having finitely many self-intersections that are endowed with under/over data. The two

Winding homology of knotoids

“…Knotoids. We review the essentials of the theory of knotoids, see [11], [5], [4] for details. A knotoid diagram in a surface Σ is an immersion K : [0, 1] → Σ having only double transversal points and over/under information for each crossing.…”

“…Knotoids are presented by knot-like diagrams that are generic immersions of the unit interval into a surface, together with the under/overpassing information at double points. Knotoids are defined as the equivalence classes of knotoid diagrams under isotopy and the Reidemeister moves, see [5] for a survey and [4] for comprehensive tables of knotoids. Intuitively, knotoids can be considered as open-ended knot-type pictures up to an appropriate equivalence.…”