On the volume conjecture for polyhedra - Département des mathématiques et applications de l'ENS PARIS Accéder directement au contenu
Article Dans Une Revue Geometriae Dedicata Année : 2015

On the volume conjecture for polyhedra

Résumé

We formulate a generalization of the volume conjecture for planar graphs. Denoting by the Kauffman bracket of the graph G whose edges are decorated by real "colors" c, the conjecture states that, under suitable conditions, certain evaluations of grow exponentially as k goes to infinity and the growth rate is the volume of a truncated hyperbolic hyperideal polyhedron whose one-skeleton is G (up to a local modification around all the vertices) and with dihedral angles given by c. We provide evidence for it, by deriving a system of recursions for the Kauffman brackets of planar graphs, generalizing the Gordon-Schulten recursion for the quantum 6j-symbols. Assuming that does grow exponentially these recursions provide differential equations for the growth rate, which are indeed satisfied by the volume (the Schlafli equation); moreover, any small perturbation of the volume function that is still a solution to these equations, is a perturbation by an additive constant. In the appendix we also provide a proof outlined elsewhere of the conjecture for an infinite family of planar graphs including the tetrahedra.

Dates et versions

hal-01940018 , version 1 (29-11-2018)

Identifiants

Citer

Francesco Costantino, François Guéritaud, Roland van Der Veen. On the volume conjecture for polyhedra. Geometriae Dedicata, 2015, ⟨10.1007/s10711-015-0086-4⟩. ⟨hal-01940018⟩
57 Consultations
0 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More