Knotoids, pseudo knotoids, Braidoids and pseudo braidoids on the Torus

Abstract

In this paper we study the theory of knotoids and braidoids and the theory of pseudo knotoids and pseudo braidoids on the torus T. In particular, we introduce the notion of {\it mixed knotoids} in $S^2$, that generalize the notion of mixed links in $S^3$, and we present an isotopy theorem for mixed knotoids. We then generalize the Kauffman bracket polynomial, $<;>$, for mixed knotoids and we present a state sum formula for $<;>$. We also introduce the notion of {\it mixed pseudo knotoids}, that is, multi-knotoids on two components with some missing crossing information. More precisely, we present an isotopy theorem for mixed pseudo knotoids and we extend the Kauffman bracket polynomial for pseudo mixed knotoids. Finally, we introduce the theories of {\it mixed braidoids} and {\it mixed pseudo braidoids} as counterpart theories of mixed knotoids and mixed pseudo knotoids respectively. With the use of the $L$-moves, that we also introduce here for mixed braidoid equivalence, we formulate and prove the analogue of the Alexander and the Markov theorems for mixed knotoids. We also formulate and prove the analogue of the Alexander theorem for mixed pseudo knotoids.