Knotoids

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.