A family of multi-variable polynomial invariants of flat virtual knots

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].