Cellular Legendrian contact homology for surfaces, part I

We give a computation of the Legendrian contact homology (LCH) DGA for an arbitrary generic Legendrian surface $L$ in the $1$-jet space of a surface. As input we require a suitable cellular decomposition of the base projection of $L$. A collection of generators is associated to each cell, and the differential is given by explicit matrix formulas. In the present article, we prove that the equivalence class of this cellular DGA does not depend on the choice of decomposition, and in the sequel [35] we use this result to show that the cellular DGA is equivalent to the usual Legendrian contact homology DGA defined via holomorphic curves. Extensions are made to allow Legendrians with non-generic cone-point singularities. We apply our approach to compute the LCH DGA for several examples including an infinite family, and to give general formulas for DGAs of front spinnings allowing for the axis of symmetry to intersect $L$.