On a weight system conjecturally related to $\mathfrak{sl}_2$

We introduce a new series $R_k$, $k=2,3,4,\dots$, of integer valued weight systems. The value of the weight system $R_k$ on a chord diagram is a signed number of cycles of even length $2k$ in the intersection graph of the diagram. We show that this value depends on the intersection graph only. We check that for small orders of the diagrams, the value of the weight system $R_k$ on a diagram of order exactly $2k$ coincides with the coefficient of $c^k$ in the value of the $\mathfrak{sl}_2$-weight system on the projection of the diagram to primitive elements.