Partial duals of plane graphs, separability and the graphs of knots

Abstract

There is a well-known way to describe a link diagram as a (signed) plane graph, called its Tait graph. This concept was recently extended, providing a way to associate a set of embedded graphs (or ribbon graphs) to a link diagram. While every plane graph arises as a Tait graph of a unique link diagram, not every embedded graph represents a link diagram. Furthermore, although a Tait graph describes a unique link diagram, the same embedded graph can represent many different link diagrams. One is then led to ask which embedded graphs represent link diagrams, and how link diagrams presented by the same embedded graphs are related to one another. Here we answer these questions by characterizing the class of embedded graphs that represent link diagrams, and then using this characterization to find a move that relates all of the link diagrams that are presented by the same set of embedded graphs.

v2: major changes

References in corpus (11)

• The Jones polynomial and graphs on surfaces
• The multivariate signed Bollobas-Riordan polynomial
• Twisted duality for embedded graphs
• Alternating sum formulae for the determinant and other link invariants
• Symmetric links and Conway sums: volume and Jones polynomial
• Partial duality and Bollobas and Riordan's ribbon graph polynomial
• On knot Floer width and Turaev genus
• Graphs on surfaces and Khovanov homology
• Unsigned state models for the Jones polynomial
• A characterization of partially dual graphs
• Non-orientable quasi-trees for the Bollobas-Riordan polynomial

Cited by in corpus (5)

• On the interplay between embedded graphs and delta-matroids
• Separability and the genus of a partial dual
• Bipartite partial duals and circuits in medial graphs
• Types of embedded graphs and their Tutte polynomials
• Inductive tools for connected ribbon graphs, delta-matroids and multimatroids