  We study Legendrian surfaces determined by cubic planar graphs. Graphs with
distinct chromatic polynomials determine surfaces that are not Legendrian
isotopic, thus giving many examples of non-isotopic Legendrian surfaces with
the same classical invariants. The Legendrians have no exact Lagrangian
fillings, but have many interesting non-exact fillings.
  We obtain these results by studying sheaves on a three-ball with microsupport
in the surface. The moduli of such sheaves has a concrete description in terms
of the graph and a beautiful embedding as a holomorphic Lagrangian submanifold
of a symplectic period domain, a Lagrangian that has appeared in the work of
Dimofte-Gabella-Goncharov [DGGo]. We exploit this structure to find conjectural
open Gromov-Witten invariants for the non-exact filling, following
Aganagic-Vafa [AV, AV2].
