The affine index polynomial invariant of flat virtual knots

Abstract

We define the affine index polynomial of a flat virtual knot in a similar way as the case of a virtual knot, and show that it is described by the affine index polynomial of any overlying virtual knot. Let K be a virtual knot, and F the underlying flat virtual knot of K. Then we have necessary conditions for the invariant of F about invertibility and amphicheirality of K and F. As applications of the invariant, we raise examples such as (1) F is non-invertible, and (2) K is non-amphicheiral. We also give an alternative proof of a fact that Hrencecin and Kauffman's flat virtual knots are mutually distinct, which is originally proved by Im, Lee and Son.