From annular to toroidal knotoids and their bracket polynomials

In this paper we study the theory of multi-knotoids of the annulus and of the torus. We present first their equivalence relation, building it up from the theory of planar knotoids to the theory of toroidal knotoids through the theory of annular knotoids. We introduce the concept of lifting annular and toroidal knotoids and examine inclusion relations arising naturally from the topology of the supporting manifolds. We also introduce the concept of mixed knotoids as special cases of planar knotoids, containing a fixed unknot for representing the thickened annulus or a fixed Hopf link for representing the thickened torus. We then extend the Turaev loop bracket for planar knotoids to bracket polynomials for annular and for toroidal knotoids, whose universal analogues recover the Kauffman bracket knotoid skein module of the thickened annulus and the thickened torus.