Tropical Fock-Goncharov coordinates for SL3-webs on surfaces I: construction

For a finite-type surface S, we study a preferred basis for the commutative algebra C[XSL3(C)(S)] of regular functions on the SL3(C)-character variety, introduced by Sikora and Westbury. These basis elements come from the trace functions associated to certain tri-valent graphs embedded in the surface S. We show that this basis can be naturally indexed by positive integer coordinates, defined by Knutson-Tao rhombus inequalities. These coordinates are related, by the geometric theory of Fock and Goncharov, to the tropical points at infinity of the dual version of the character variety.