Knots Related by Knotoids

Turaev recently introduced the concept of a knotoid as a particular sort of knotted arc. There are several maps from knotoids to ordinary knots, or knotted circles. The two most natural ways of defining such maps give us an interesting relation between pairs of knots. In this paper, we explore this relation and ask the question: which pairs of knots are related to each other by a knotoid?.