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

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.

31 pages, 11 figures

References in corpus (8)

• The Jones polynomial and graphs on surfaces
• The multivariate signed Bollobas-Riordan polynomial
• The Kauffman bracket of virtual links and the Bollobás-Riordan polynomial
• Knot invariants and the Bollobas-Riordan polynomial of embedded graphs
• Alternating sum formulae for the determinant and other link invariants
• Partial duality and Bollobas and Riordan's ribbon graph polynomial
• A characterization of partially dual graphs
• Expansions for the Bollobas-Riordan polynomial of separable ribbon graphs

Cited by in corpus (8)

• Separability and the genus of a partial dual
• Matroids, Delta-matroids and Embedded Graphs
• Bipartite partial duals and circuits in medial graphs
• Partial duals of plane graphs, separability and the graphs of knots
• On the Seifert graphs of a link diagram and its parallels
• Types of embedded graphs and their Tutte polynomials
• Topological Tutte Polynomial
• Excluded minors and the ribbon graphs of knots