The space of framed chord diagrams as a Hopf module

Karev, Maksim

doi:10.1142/s0218216515500145

Cited by 5 publications

(9 citation statements)

References 7 publications

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“…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. )…”

Section: 3mentioning

confidence: 99%

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Ribbon graphs and bialgebra of Lagrangian subspaces

Kleptsyn

Smirnov

2016

J. Knot Theory Ramifications

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Abstract. To each ribbon graph we assign a so-called L-space, which is a Lagrangian subspace in an even-dimensional vector space with the standard symplectic form. This invariant generalizes the notion of the intersection matrix of a chord diagram. Moreover, the actions of Morse perestroikas (or taking a partial dual) and Vassiliev moves on ribbon graphs are reinterpreted nicely in the language of L-spaces, becoming changes of bases in this vector space. Finally, we define a bialgebra structure on the span of L-spaces, which is analogous to the 4-bialgebra structure on chord diagrams.

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Section: 3mentioning

confidence: 99%

“…[13]. Recently M. Karev [14] showed that this space is a module over the algebra of unframed chord diagrams.…”

Section: Introductionmentioning

confidence: 99%

Ribbon graphs and bialgebra of Lagrangian subspaces

Kleptsyn

Smirnov

2016

J. Knot Theory Ramifications

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“…We can consider chord diagrams only with framings 0 and the corresponding 4T-relation. As a result we obtain the submodule M of M f , see [12].…”

Section: Framed Chord Diagramsmentioning

confidence: 99%

“…a connected sum, and the coproduct is given by summing up the tensor products of pairs of chord diagrams formed by a decomposition of the set of chords into two complimentary subsets. Analogously, the space A f of framed chord diagrams is endowed with the comultiplication transforming it to the coassociative cocommutative coalgebra [12,13].…”

Section: Framed Chord Diagramsmentioning

confidence: 99%

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A parity map of framed chord diagrams

Ilyutko

Manturov

2015

J. Knot Theory Ramifications

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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.

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