Values of the $\mathfrak{sl}_2$ weight system on a family of graphs that are not the intersection graphs of chord diagrams

The Chmutov-Lando theorem claims that the value of a weight system (a function on the chord diagrams that satisfies the four-term Vassiliev relations) corresponding to the Lie algebra $\mathfrak{sl}_2$ depends only on the intersection graph of the chord diagram. We compute the values of the $\mathfrak{sl}_2$ weight system at the graphs in several infinite series, which are the joins of a graph with a small number of vertices and a discrete graph. In particular, we calculate these values for a series in which the initial graph is the cycle on five vertices; the graphs in this series, apart from the initial one, are not intersection graphs. We also derive a formula for the generating functions of the projections of graphs equal to the joins of an arbitrary graph and a discrete graph to the subspace of primitive elements of the Hopf algebra of graphs. Using the formula thus obtained, we calculate the values of the $\mathfrak{sl}_2$ weight system at projections of the graphs of the indicated form onto the subspace of primitive elements. Our calculations confirm Lando's conjecture concerning the values of the $\mathfrak{sl}_2$ weight system at projections onto the subspace of primitives. Bibliography: 17 titles.

chord diagram, weight system, intersection graph, join of graphs, Hopf algebra