  In this paper, we construct quantum invariants for knotoid diagrams in
$\mathbb{R}^2$. The diagrams are arranged with respect to a given direction in
the plane ({\it Morse knotoids}). A Morse knotoid diagram can be decomposed
into basic elementary diagrams each of which is associated to a matrix that
yields solutions of the quantum Yang-Baxter equation. We recover the bracket
polynomial, and define the rotational bracket polynomial, the binary bracket
polynomial, the Alexander polynomial, the generalized Alexander polynomial and
an infinity of specializations of the Homflypt polynomial for Morse knotoids
via quantum state sum models.
