Non-orientable quasi-trees for the Bollobas-Riordan polynomial

TL;DR

This paper extends the quasi-tree expansion to non-orientable ribbon graphs, explores duality properties of the Bollobas-Riordan polynomial, and derives a connected state expansion of the Kauffman bracket for virtual links.

Contribution

It introduces a generalized quasi-tree expansion for non-orientable ribbon graphs and links it to virtual link invariants, expanding the theoretical framework of graph polynomials.

Findings

Extended quasi-tree expansion to non-orientable graphs

Established duality properties of the Bollobas-Riordan polynomial

Derived a connected state expansion of the Kauffman bracket

Abstract

We extend the quasi-tree expansion of A. Champanerkar, I. Kofman, and N. Stoltzfus to not necessarily orientable ribbon graphs. We study the duality properties of the Bollobas-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 S. Chmutov.