Knotoids

TL;DR

This paper introduces the concept of knotoids, a generalization of knots represented by diagrams with endpoints, and explores their properties, algebraic structures, and potential applications in knot theory.

Contribution

It presents the formal definition of knotoids, studies their algebraic structure, and discusses their relation to classical knots and invariants.

Findings

Knotoids generalize classical knots in $S^3$.

The semigroup of knotoids is analyzed.

Applications to knot invariants are discussed.

Abstract

We introduce and study knotoids. Knotoids are represented by diagrams in a surface which differ from the usual knot diagrams in that the underlying curve is a segment rather than a circle. Knotoid diagrams are considered up to Reidemeister moves applied away from the endpoints of the underlying segment. We show that knotoids in $S^2$ generalize knots in $S^3$ and study the semigroup of knotoids. We also discuss applications to knots and invariants of knotoids.