Pseudo links in handlebodies

In this paper we introduce and study the theory of pseudo links in the genus $g$ handlebody, $H_g$. Pseudo links are links with some missing crossing information that naturally generalize the notion of knot diagrams. The motivation for studying these relatively new knotted objects lies in the fact that pseudo links may be used to model DNA knots, since it is not uncommon for biologists to obtain DNA knots for which it is not possible to tell a positive from a negative crossing. We consider pseudo links in $H_g$ as mixed pseudo links in $S^3$ and we generalize the Kauffman bracket polynomial for the category of pseudo links in $H_g$. We then pass on the category of mixed pseudo braids, that is, pseudo braids whose closures are pseudo links in $H_g$, and we formulate the analogue of the Alexander theorem. It is worth mentioning that the theory of pseudo links is close related to the theory of singular links and that all results in this paper may be used for studying singular links in $H_g$.