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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. In the study of the sl2 weight system in [8], it was shown
that its value on a chord diagram depends on the intersection graph of the
diagram rather than on the diagram itself. Moreover, it was shown that
the value of this weight system on an intersection graph depends on the cy-
cle matroid of the graph rather than on the graph itself. This result arose
the question whether there is a natural way to introduce a 4-term relation
on the space spanned by matroids, similar to the one for graphs [13]. It
happened however that the answer is negative: there are graphs having iso-
morphic cycle matroids such that applying the “second Vassiliev move” to a
pair of corresponding vertices a, b of the graphs we obtain two graphs with
nonisomorphic matroids.
The goal of the present paper is to show that the situation is different for
binary delta-matroids: 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. Moreover, this mapping admits
a natural extension to chord diagrams on several circles, which correspond
to singular links. Delta-matroids were introduced by A. Bouch´et [4] for the
purpose of studying embedded graphs, whence their relationship with (mul-
tiloop) chord diagrams is by no means unexpected. Some evidence for the
existence of such a relationship can be found, for example, in [2], where the
Tutte polynomial for embedded graphs has been introduced. The authors
show that this polynomial depends on the delta-matroid of the embedded
graph rather than the graph itself and satisfies the Vassilev 4-term relation.