An index polynomial invariant for flat virtual knots

We introduce a polynomial invariant of flat virtual knots which is sometimes useful for determining whether given flat virtual knots are invertible or not, and for finding the virtual crossing number of flat virtual knots. Also we give several properties of the polynomial invariant for flat virtual knots and examples. In particular, we show that the conjecture given by Hrencecin and Kauffman (2003) [8] is true and so an infinite class of flat virtual knots {Un}(n=1,2,…) are mutually distinct.