  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
$H_g$.
