A parity map of framed chord diagrams

Авторы: Denis Petrovich Ilyutko, Vassily Olegovich Manturov
Журнал: Journal of Knot Theory and its Ramifications
Том: 24
Номер: 13
Год издания: 2015
Издательство: World Scientific Publishing Co
Местоположение издательства: Singapore
DOI: 10.1142/S0218216515410060
Аннотация: 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 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 ℳ2 which is generated by chord diagrams on two circles and factored by 4T-relations. Then we construct a "parity" map from the module of framed chord diagrams into ℳ2 and a weight system on ℳ2. 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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