Delta-matroids and Vassiliev invariants

TL;DR

This paper explores the relationship between delta-matroids and Vassiliev invariants, demonstrating that delta-matroids can be used to define 4-term relations compatible with chord diagrams, unlike cycle matroids.

Contribution

It introduces a 4-term relation for binary delta-matroids that aligns with chord diagram invariants, extending the framework for studying Vassiliev invariants.

Findings

Delta-matroids admit 4-term relations compatible with chord diagrams.

The mapping from chord diagrams to delta-matroids respects 4-term relations.

Extension of the delta-matroid framework to chord diagrams on multiple circles.

Abstract

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, 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. It happened however that the answer is negative: there are graphs having isomorphic 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. Bouchet for the purpose of studying embedded graphs, whence their relationship with (multiloop) chord diagrams is by no means unexpected. Some evidence for the existence of such a relationship can be found, for example, the Tutte polynomial for embedded graphs has been introduced. It was shown that this polynomial depends on the delta-matroid of the embedded graph rather than the graph itself and satises the Vassilev 4-term relation.