Linking number for graph-links and some classification of graph-links

In [Ilyutko and Safina, Graph-links: nonrealizability, orientation, and Jones polynomial, J. Math. Sc.214(5) (2016) 632–664], the first named author defined the notion of an oriented graph-link and constructed a writhe number of a vertex of an oriented graph-link, which equals the “real” writhe number of a crossing in the realizable case. As a result the Jones polynomial was defined for oriented graph-links and the first example of a non-realizable graph-link with more than one component was found. Despite the fact that all necessary definitions were given the authors did not define the notion of linking number for graph-links. In this paper, we are eliminating this deficiency. Moreover, we classify all graph-links having a representative with less than $4$ vertices.