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Moscow Mathematical Journal

Volume 17, Issue 4, October–December 2017  pp. 741–755.

Delta-Matroids and Vassiliev Invariants

Authors:  Sergey Lando (1) and Vyacheslav Zhukov (2)
Author institution:(1) National Research University Higher School of Economics, Skolkovo Institute of Science and Technology
(2) National Research University Higher School of Economics


Vassiliev (finite type) invariants of knots can be described in terms of weight systems. These are functions on chord diagrams satisfying so-called 4-term relations. The goal of the present paper is to show that one can define both the first and the second Vassiliev moves for binary delta-matroids and introduce a 4-term relation for them in such a way that the mapping taking a chord diagram to its delta-matroid respects the corresponding 4-term relations.

Understanding how the 4-term relation can be written out for arbitrary binary delta-matroids motivates introduction of the graded Hopf algebra of binary delta-matroids modulo the 4-term relations so that the mapping taking a chord diagram to its delta-matroid extends to a morphism of Hopf algebras. One can hope that studying this Hopf algebra will allow one to clarify the structure of the Hopf algebra of weight systems, in particular, to find reasonable new estimates for the dimensions of the spaces of weight systems of given degree.

2010 Math. Subj. Class. 05C31, 57M27.

Keywords: Delta-matroid, binary delta-matroid, finite order knot invariants, chord diagram, weight system, 4-term relations.

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