An invariant of graph-links valued in graphs and an almost classification of graph-links

Abstract

In [V. O. Manturov, An almost classification of free knots, Dokl. Math. 88(2) (2013) 556–558.] the second author constructed an invariant which in some sense generalizes the quantum [Formula: see text] link invariant of Kuperberg to the case of free links. In this paper, we generalize this construction to free graph-links. As a result, we obtain an invariant of free graph-links with values in linear combinations of graphs. The main property of this invariant is that under certain conditions on the representative of the free graph-link, we can recover this representative from the value invariant on it. In addition, this invariant allows one to partially classify free graph-links.