On the height of knotoids

Abstract

Knotoid diagrams are defined in analogy to open ended knot diagrams with two distinct 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 (Kauffman, J Knot Theory Ramif 21(3), 37, 2012) [6], (Kauffman, J Knot Theory Ramif 22(4), 30, 2013) [8], respectively, but here are described entirely in terms of knotoids in S2. We reprise here our results given in (Gügümcü, Kauffman, Eur J Combin 65C, 186–229, 2017) [3] that show that both polynomials give a lower bound for the height of knotoids.