A parity map of framed chord diagrams
Denis Ilyutko, Vassily Manturov

TL;DR
This paper introduces a new module and a covering map for framed chord diagrams, demonstrating the absence of a natural connected sum operation and exploring related linear diagrams.
Contribution
It defines a new module $ ext{M}_2$, constructs a covering map and weight system, and shows that a natural connected sum operation is not well-defined for framed chord diagrams.
Findings
Connected sum is not well-defined for framed chord diagrams.
A new module $ ext{M}_2$ is introduced and related to framed chord diagrams.
The paper explores linear diagrams as a related concept.
Abstract
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 which is generated by chord diagrams on two circles and factored by T-relations. Then we construct a "covering" map from the module of framed chord diagrams into and a weight system on . 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.
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