Chapter 1 On the height of knotoids Neslihan G ¨ ug¨ umc¨ u, Louis H. Kauffman Abstract Knotoid diagrams are defined in analogy to open ended knot diagrams with two distict endpoints that can be located in any region of the diagram. The height of a knotoid is the minimal crossing distance between the endpoints taken over all equivalent knotoid diagrams. We define two knotoid invariants; the affine index polynomial and the arrow polynomial that were originally defined as virtual knot invariants given in [3,9], respectively, but here are described entirely in terms of knotoids in S 2 . We reprise here our results given in [4] that show that both poly- nomials give a lower bound for the height of knotoids. 1.1 Introduction The theory of knotoids was introduced by V. Turaev [17] in 2012. A knotoid diagram [17] is an open ended knot diagram with two endpoints that can be located in any region of the diagram. The theory of knotoids forms a new diagrammatic theory that is an extension of the classical knot theory. In this paper, we give an exposition of two new polynomial knotoid invariants that were constructed in [4]. It is natural to examine knotoids in the context of virtual knot theory [6,7]. Virtual knots are knots in thickened surfaces (or knot diagrams in surfaces) taken up to handle stabilization. There is a diagrammatic theory for virtual knots, as we explain briefly in this paper. The endpoints of a knotoid diagram can be connected to form what we call the virtual closure of the diagram. This way of connecting the endpoints of a knotoid diagram forms a well-defined map from the set of knotoids to the set of virtual knots. Virtual knot invariants can be then applied to extract knotoid invariants by using the virtual closure map. Louis H. Kauffman University of Illinois at Chicago, USA, e-mail: kauffman@uic.edu Neslihan G ¨ ug¨ umc¨ u National Technical University of Athens, Greece e-mail: nesli@central.ntua.gr 1