Values of the $\mathfrak{sl}_2$ weight system on the chord diagrams whose intersection graphs are complete graphs.

A weight system is a function on the chord diagrams that satisfies Vassiliev's $4$-term relation. Using the Lie algebra ${sl}_2$ we can construct the simplest nontrivial weight system. The resulting ${sl}_2$ weight system takes values in the space of polynomials of one variable and is completely determined by the Chmutov-Varchenko recurrence relations.
Although the definition of the ${sl}_2$ weight system is rather simple, calculations of its values are laborious, and therefore concrete values are only known for a small number of chord diagrams. As concerns the explicit form of values at chord diagrams with complete intersection graphs, Lando stated a conjecture, which initially could only be proved for the coefficients at linear terms of the values of the weight system. We prove this conjecture in full using the Chmutov-Varchenko recurrence relations and the linear operators we introduce for adding a chord to a share, which is the subset of chords of the diagram with endpoints on two selected arcs. Also, relying on the generating function of the values of the ${sl}_2$ weight system at chord diagrams with complete intersection graphs, we prove that the quotient space of shares modulo the recurrence relations is isomorphic to the space of polynomials in two variables.