Arrow ribbon graphs

Understand

We introduce an additional structure on ribbon graphs, arrow structure.

• We extend the Bollob\'as-Riordan polynomial to ribbon graph with this structure.
• The extended polynomial satisfies the contraction-deletion relations and naturally behaves with respect to the partial duality of ribbon graphs.
• We construct an arrow ribbon graph from a virtual link whose extended Bollob\'as-Riordan polynomial specializes to the arrow polynomial of the virtual link recently introduced by H.Dye and L.Kauffman.