Non-orientable quasi-trees for the Bollobás–Riordan polynomial

We extend the quasi-tree expansion of Champanerkar et al. (2007) [2] to not necessarily orientable ribbon graphs. We study the duality properties of the Bollobás–Riordan polynomial in terms of this expansion. As a corollary, we get a “connected state” expansion of the Kauffman bracket of virtual link diagrams. Our proofs use extensively the partial duality of Chmutov (2009) [3].