New invariants of knotoids

In this paper we construct new invariants of knotoids including the odd writhe, the parity bracket polynomial, the affine index polynomial and the arrow polynomial, and give an introduction to the theory of virtual knotoids. The invariants in this paper are defined for both classical and virtual knotoids in analogy to corresponding invariants of virtual knots. We show that knotoids in S2 have symmetric affine index polynomials. The affine index polynomial and the arrow polynomial provide bounds on the height (minimum crossing distance between endpoints) of a knotoid in S2.