On values of \(\mathfrak{sl}_3\) weight system on chord diagrams whose intersection graph is complete bipartite

Abstract

Each knot invariant can be extended to singular knots according to the skein rule. A Vassiliev invariant of order at most \(n\) is defined as a knot invariant that vanishes identically on knots with more than \(n\) double points. A chord diagram encodes the order of double points along a singular knot. A Vassiliev invariant of order \(n\) gives rise to a function on chord diagrams with \(n\) chords. Such a function should satisfy some conditions in order to come from a Vassiliev invariant. A weight system is a function on chord diagrams that satisfies so-called 4-term relations. Given a Lie algebra \(\mathfrak{g}\) equipped with a non-degenerate invariant bilinear form, one can construct a weight system with values in the center of the universal enveloping algebra \(U(\mathfrak{g})\). In this paper, we calculate \(\mathfrak{sl}_3\) weight system for chord diagram whose intersection graph is complete bipartite graph \(K_{2,n}\).