On a weight system conjecturally related to sl2sl2

We introduce a new series RkRk, k=2,3,4,…k=2,3,4,…, of integer valued weight systems. The value of the weight system RkRk on a chord diagram is a signed number of cycles of even length 2k2k 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 RkRk on a diagram of order exactly 2k2k coincides with the coefficient of ckck in the value of the sl2sl2-weight system on the projection of the diagram to primitive elements.