A polynomial invariant for knotoids

A polynomial invariant for multi-knotoids in $S^2$ is given by an elementary and combinatorial method. It is shown that the invariant is an extension of ``a'' HOMFLY polynomial for multi-knotoids and that there exist infinitely many non-trivial knotoids with trivial HOMFLY polynomial. Furthermore, formulas between the polynomials for a given multi-knotoid, its mirror and reverse images are given. By using the formulas, it is revealed that each knotoid with less than four crossings is invertible.