Deltamatroids and Vassiliev invariants
Abstract
Vassiliev (finite type) invariants of knots can be described in terms of weight systems. These are functions on chord diagrams satisfying socalled 4term 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 4term 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 deltamatroids: one can define both the first and the second Vassiliev moves for binary deltamatroids and introduce a 4term relation for them in such a way that the mapping taking a chord diagram to its deltamatroid respects the corresponding 4term relations. Moreover, this mapping admits a natural extension to chord diagrams on several circles, which correspond to singular links. Deltamatroids 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 deltamatroid of the embedded graph rather than the graph itself and satises the Vassilev 4term relation.
 Publication:

arXiv eprints
 Pub Date:
 January 2016
 arXiv:
 arXiv:1602.00027
 Bibcode:
 2016arXiv160200027L
 Keywords:

 Mathematics  Combinatorics;
 05C10
 EPrint:
 20 pages