Volume Bounds for Trivalent Planar Graphs

Abstract

This paper studies the volumes of hyperbolic planar trivalent graphs. The connection between fully augmented links and trivalent planar graphs allows us to apply previous work on knots and links to this setting. Exact volume bounds for graphs with up to 12 vertices are calculated. Purcell’s sharp lower bound for the volumes of fully augmented links translates directly to a lower bound for trivalent graphs. Moreover, a natural cell decomposition on trivalent graphs allows us to interpret upper bounds for link volumes in the graph theoretic setting. This is done for both the Agol Thurston tetrahedral upper bound and Adams' bipyramidal upper bound. An infinite family of graphs, inspired by Agol-Thurston’s infinite chain link fence, proves these bounds are asymptotically equivalent.