Distinguished bases in the wrapped Floer cohomology of tropical Lagrangian surfaces

We introduce a new construction for tropical Lagrangian surfaces in $({\mathbb{C}}^{\ast }{)}^2$. This construction makes the surfaces special Lagrangian, which gives a strong control over the asymptotic behavior of holomorphic disks near each cylindrical end. As a result, we find a distinguished basis of the wrapped Floer cohomology ring $H{W}^{\bullet }(L_f,L_f)$ for several different examples and find the corresponding distinguished bases of functions on the mirror curves.