  As a generalization of the classical knots, knotoids deal with the open ended
knot diagrams in a surface. In recent years, many polynomial invariants for
knotoids have appeared, such as the bracket polynomial, the index polynomial
and the $n$th polynomial, etc. In this paper, we introduce a new polynomial
invariant $F$-polynomial for knotoids and discuss some properties of the
$F$-polynomial. Then, we construct a family of knotoid diagrams which can be
distinguished from each other by the $F$-polynomial but cannnot be
distinguished by the index polynomial and the $n$th polynomial.
