A parity map of framed chord diagrams

Abstract

We consider framed chord diagrams, i.e. chord diagrams with chords of two types. It is well known that chord diagrams modulo [Formula: see text]T-relations admit a 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 this paper, we first define a new module [Formula: see text] which is generated by chord diagrams on two circles and factored by [Formula: see text]T-relations. Then we construct a “parity” map from the module of framed chord diagrams into [Formula: see text] and a weight system on [Formula: see text]. 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.