Schemes of involving dual variables in inverse barrier functions for problems of linear and convex programming

With any stable map from a 3-manifold to ℝ³, we associate a graph with weights in its vertices and edges. These graphs are A-invariants from a global viewpoint. We study their properties and show that any tree with zero weights in its vertices and aleatory weights in its edges can be the graph of a stable map from S ³ to ℝ³.