A polynomial invariant and the forbidden move of virtual knots

1 Preliminaries A virtual knot diagram is presented by a knot diagram having virtual crossings as well as real crossings in Fig. 1. Two virtual knot diagrams are equivalent if one can be obtained from the other by a finite sequence of generalized Reidemeister moves in Fig. 2. The equivalence class of virtual knot diagrams modulo the generalized Reidemeister moves is called a virtual knot. Virtual knots can be described by Gauss diagrams. The Gauss diagram of a virtual knot is an oriented circle as the preimage of the immersed circle with chords connecting the