Knotoids
In this paper, the author introduces the concept of \textit{knotoids}, which are represented by diagrams in a surface with the underlying curve a segment, rather than a circle. Knotoids are defined as equivalence classes of knotoid diagrams modulo the usual Reidemeister moves applied away from the endpoints. The author shows that knotoids in \(S^2\) generalize knots in \(S^3\) and defines a semigroup on knotoids in \(S^2\) that contains the usual semigroup of knots in the center. He uses techniques from the study of theta-curves to establish properties of this semigroup.NEWLINENEWLINEEvery knot in \(S^3\) may be represented by a knotoid diagram and every knotoid diagram in \(S^2\) determines a unique knot in \(S^3\) (actually two knots, see the paper for more details). Thus, studying knotoids may lead to more information about knots. For example, a Seifert surface can be constructed from a knotoid diagram in \(S^2\), and the author is able to use this construction to determine an upper bound on the genus of the associated knot. This upper bound is sometimes stronger than the usual upper bound on genus obtained from constructing the Seifert surface of the knot diagram.NEWLINENEWLINEThe paper also discusses an extension of several knot invariants to knotoids, including the knot group, crossing number, and the bracket polynomial. The author also introduces a skein algebra of knotoids in \(S^2\). In the final section, the author describes a surjective map between knotoids in \({\mathbb R}^2\) and knotoids in \(S^2\). He ends by giving examples of knotoids that are non-trivial in \({\mathbb R}^2\) but whose images are trivial in \(S^2\).
scientific article; zbMATH DE number 6029033
scientific article