Generalized duality for graphs on surfaces and the signed Bollobás–Riordan polynomial

Abstract

We generalize the natural duality of graphs embedded into a surface to a duality with respect to a subset of edges. The dual graph might be embedded into a different surface. We prove a relation between the signed Bollobás-Riordan polynomials of dual graphs. This relation unifies various recent results expressing the Jones polynomial of links as specializations of the Bollobás-Riordan polynomials.

Parametric representation of rank d tensorial group field theory: Abelian models with kinetic term ∑sps+μ

“…These graphs are quite well understood and still intensively investigated. For a complete definition of ribbon graphs, we will refer to one of the following standard references [71,72,73,74,75] (the last reference offers an up-to-date survey). The case of ribbon graphs with half-edges or half-ribbons and their relation to Physics, the work by Krajewski and co-workers [69] is seminal.…”

“…For the GW model in 4D, the polynomials on ribbon graphs discovered in the parametric representation of this model [82] were deformed versions of the Bollobàs-Riordan polynomial [73] [72]. The recurrence relation obeyed by these invariants is however much more involved [69] (a four-term recurrence using Chmutov partial duality [74]). Our remaining task is to investigate the types of relations which are satisfied by the identified functions U od/ev G andW G (W G will satisfy relations which can be inferred from these points).…”

“…where G e stands for the so-called Chmutov partial dual [74] of G with respect to the edge e. This operation can simply be explained as follows: one must cut all lines in G except e, then perform a dual on the pinched graphG, and glue black all edges previously cut. The interest of introducing such partial dual reflects on the contraction operation: G/e = G e − e. It turns out that the GW model can be expressed as well as a matrix model [79].…”

Partial duality and closed 2-cell embeddings

“…No one, however, seems to have considered a partial version of geometric duality for graph embeddings, until Chmutov [6] in 2009. He was motivated the desire to unify several Thistlethwaite-type theorems; results of this type relate polynomials for knots or links to graph polynomials.…”

“…If we start with a closed 2-cell graph embedding, taking the partial dual of any single edge (necessarily a link, or non-loop) creates a loop, and a graph embedding with a loop and at least one other edge is not closed 2-cell. (See [6,Subsection 1.7] or [12, pp. 32-33].…”