Arrow ribbon graphs

TL;DR

This paper introduces arrow ribbon graphs, extends the Bollobás-Riordan polynomial to include arrow structures, and connects these concepts to virtual link invariants, generalizing classical theorems.

Contribution

It develops the theory of arrow ribbon graphs and extends the Bollobás-Riordan polynomial to incorporate arrow structures, linking it to virtual link invariants.

Findings

Extended Bollobás-Riordan polynomial satisfies contraction-deletion relations.

The polynomial behaves naturally under partial duality of ribbon graphs.

Generalizes Thistlethwaite's theorem to the arrow polynomial of virtual links.

Abstract

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. This result generalizes the classical Thistlethwaite theorem to the arrow polynomial of virtual links.