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.
Citation
Vladimir Turaev. "Knotoids." Osaka J. Math. 49 (1) 195 - 223, March 2012.
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