Moscow Mathematical Journal
Volume 17, Issue 4, October–December 2017 pp. 741–755.
Delta-Matroids and Vassiliev Invariants
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.
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
Summary:
Keywords: Delta-matroid, binary delta-matroid, finite order knot invariants, chord diagram, weight system, 4-term relations.
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