Shadow formula for the Vassiliev invariant of degree two

Abstract

AbstractIn this paper I present the Vassiliev invariant of degree 2 of a knot as a polynomial of degree 4 in gleams of a shadow presenting the knot. The coefficients of this polynomial involve Strangeness (a numerical characteristic of a generic immersion of the circle into the plane introduced by Arnold [1]) and a spherical index. The latter is a characteristic of a generic immersed circle on the sphere with three holes, which is invariant with respect to homotopy. I define it by an explicit combinatorial formula