A parity map of framed chord diagrams

We consider framed chord diagrams, i.e. chord diagrams with chords of two types. It is well known that chord diagrams modulo 4T-relations admit Hopf algebra structure, where the multiplication is given by any connected sum with respect to the orientation. But in the case of framed chord diagrams a natural way to define a multiplication is not known yet. In the present paper, we first define a new module $\mathcal{M}_2$ which is generated by chord diagrams on two circles and factored by $4$T-relations. Then we construct a "covering" map from the module of framed chord diagrams into $\mathcal{M}_2$ and a weight system on $\mathcal{M}_2$. Using the map and weight system we show that a connected sum for framed chord diagrams is not a well-defined operation. In the end of the paper we touch linear diagrams, the circle replaced by a directed line.