The writhe and first intersection polynomials of virtual knots of supporting genus one

Abstract

The supporting genus of a virtual knot is the minimal genus of all thickened surfaces where the virtual knot lies. In this paper, we study several polynomial invariants of a virtual knot of supporting genus one. We show that the first intersection polynomial of any virtual knot of supporting genus one is reciprocal. We also give a characterization of the writhe polynomial of a virtual knot of supporting genus one.