We construct a new polynomial invariant of flat virtual knots by using the Formula: see text–polynomial introduced in an earlier work of the author. In defining the Formula: see text-polynomial, we assign a monomial Formula: see text to each crossing Formula: see text of a virtual knot diagram Formula: see text. By modifying Formula: see text suitably, we define a polynomial for flat virtual knots. We show that Formula: see text is invariant under the flat Reidemeister moves, and hence is a flat virtual knot invariant. Furthermore, explicit computations demonstrate that the new invariant distinguishes flat virtual knots that are invisible to the Kauffman–Richter polynomial Formula: see text. We show that if Formula: see text then Formula: see text. Moreover, we generalize the polynomial Formula: see text to Formula: see text for each natural number Formula: see text.
Myeong–Ju Jeong (Fri,) studied this question.