On the Vassiliev invariants of degree \(\leq 3\)
This article aims to give an elementary account of Vassiliev invariants of a knot with degree 2 or 3. The linear space of invariants of degree 2 has dimension 1, as does the space of invariants of degree 3 modulo those of lower degree. In the first part of the paper explicit formulae for the value of an invariant of degree 2 on a singular knot with two nodes is found in terms of its value on the trefoil and the standard weight function on 2-node chord diagrams, while a similar formula for a degree 3 invariant on a 3- node knot also involves the value of the invariant of the figure-eight knot. The author goes on to construct an invariant \(V_ 2(K)\) of a knot \(K\) by a consideration of all knots with the same given projection as \(K\). The invariant \(V_ 2\) is defined as a sum over pairs of nodes in the projection weighted by the standard weight function for the corresponding diagram and involving a measure of the difference over the two chosen nodes between \(K\) and a fixed descending knot diagram with the same projection. It is shown from first principles that \(V_ 2\) is an invariant of \(K\) itself, and that any Vassiliev invariant \(f\) of degree 2 satisfies \(f(K)= V_ 2(K) f(T)\), where \(T\) is the trefoil knot. It follows that \(V_ 2\) is itself a Vassiliev invariant of degree 2, and indeed is exactly the coefficient of \(z^ 2\) in the Conway polynomial. The construction given here is presented in the same way as previous work by the author in which he shows how to calculate the Kervaire (Arf) invariant of a knot in terms of pairs of double points of a projection of the knot. A similar calculation for invariants of degree 3 is then given, based on sums over triples of nodes in a knot projection. These are finally related explicitly to power series coefficients in the Jones polynomial.
scientific article; zbMATH DE number 597695