From the introduction: There is a method for encoding classical links by means of Gauss diagrams and Reidemeister moves. However, not all Gauss diagrams determine diagrams of classical links; the passage to arbitrary Gauss diagrams gives rise to the theory of virtual links.
Virtual diagrams can also be described by chord diagrams constructed by using rotating circuits. In turn, chord diagrams are described by their intersection graphs, but not all graphs originate from chord diagrams. Thus, virtual diagrams are determined (with some information loss) by graphs. An analogy arises: the passage from realizable Gauss diagrams (classical knots) to arbitrary chord diagrams leads to the concept of a virtual knot, and the passage from realizable (by means of chord diagrams) graphs to arbitrary graphs leads to the concept of a new object, graph-link, which we introduce in this paper.
In [L. Traldi and L. Zulli, J. Knot Theory Ramifications 18, No. 12, 1681–1709 (2009; Zbl 1204.57009)], similar objects for knot diagrams were constructed by applying Gauss diagrams; we use a different approach, which is based on rotating circuits [see V. O. Manturov, Banagl, Markus (ed.) et al., The mathematics of knots. Theory and application. Berlin: Springer (ISBN 978-3-642-15636-6/hbk; 978-3-642-15637-3/ebook). Contributions in Mathematical and Computational Sciences 1, 169–197 (2011; Zbl 1219.05047)]. The approach employing rotating circuits makes it possible to encode not only knot diagrams but also diagrams of links with any number of components. For graph-links, we construct a Jones polynomial and prove a minimality theorem.