  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.
