THE PARITY WRITHE POLYNOMIALS FOR VIRTUAL KNOTS AND FLAT VIRTUAL KNOTS

Abstract

We introduce a polynomial invariant with two variables for an oriented virtual knot, which refines the odd writhe polynomial with one variable due to Cheng by using a modified version of the warping degree. Our invariant is a Vassiliev invariant of degree one, reduces to one variable for a checkerboard colorable virtual knot, vanishes for classical knots, and detects non-invertibility and non-amphicheirality for some cases. We raise some examples to show effectiveness of our invariant. Moreover we define a similar invariant for a flat virtual knot.