The Topological Background of Schwarzite Physics

About 10 years ago the synthesis of random carbon schwarzites by supersonic cluster beam deposition has endowed the rich sp(2) carbon family with its three-dimensional member. Its reluctance to grow as a three-periodic minimal surface according to topological and physical predictions still prevents schwarzites from being a hot topic, although spongy carbon is already having countless applications. Understanding the links between topology and quantum structure, possibly with the help of large-scale quantum molecular dynamics simulations should trace the route to the synthesis of periodic schwarzites. In this perspective, after a brief account on the growth and characterization of spongy carbon, we review the elementary topology of schwarzites, their stability and growth conditions as derived from pure topological arguments, the electronic structure and the electron-phonon interaction of the smallest periodic schwarzites and what can be learnt by the topological monitoring of quantum molecular dynamics.