We introduce a new combinatorial encoding of 3-manifolds by planar graphs and discuss its interrelations to other known descriptions of 3-manifolds, electric networks, and knots. This graph encoding has an interesting algebraic structure (being related to Lie algebras and braided ribbon Hopf algebras) and seems to be well-suited for computation of perturbative invariants. While in the usual setup such invariants are given by complicated Feynman integrals over configuration spaces, in this case they turn into a simple weighted count of certain subgraphs. We describe simplest invariants obtained in this way and propose a general theory of finite type invariants                                                                                                                                                                                 
 

https://indico.math.cnrs.fr/event/15077/