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 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 M 2 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 M 2 and a weight system on 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.

Ribbon graphs and bialgebra of Lagrangian subspaces

“…However (see the discussion in [15]), for a certain time it was supposed that the framed chord diagrams modulo these relations do not form a bialgebra. And indeed, it was shown recently by Ilyutko and Manturov in [13]. In a similar way each multicomponent plane curve with self-tangencies gives rise to a ribbon graph.…”

“…Unfortunately, this fails even on the stage of framed chord diagrams (one-vertex ribbon graphs, not necessarily orientable), modulo the four-term relation. As it was discussed in [15] and shown in [13], the attempt to multiply framed chord diagrams in a similar way as in the previous subsection fails: this multiplication is not well-defined. (Recently M. Karev [14] showed that framed chord diagrams form a module over the bialgebra of chord diagrams.…”

“…It admits a coalgebra structure, but the multiplication of framed chord diagrams is not well-defined, cf. [13]. Recently M. Karev [14] showed that this space is a module over the algebra of unframed chord diagrams.…”

Delta-Matroids and Vassiliev Invariants

“…It was asked in [14] whether imposing the 4-term relations allows one to define multiplication on framed chord diagrams as well. Recently, D. P. Ilyutko and V. O. Manturov [10] answered this question in negative. The results of the present section show, however, that on the level of (binary) delta-matroids we obtain Hopf algebra structures not only for framed chord diagrams, but for arbitrary embedded graphs as well.…”

Partial-dual polynomial as a framed weight system

“…2 Framed chord diagrams Definition 2.1 ( [8,10]). A chord diagram is a cubic graph consisting of a selected oriented Hamiltonian cycle (the core circle) and several non-oriented edges (chords) connecting points on the core circle.…”