Contact homology computations for singular Legendrian knots and the surgery formula in two dimensions

The Chekanov-Eliashberg dg-algebra is an algebraic invariant of Legendrian submanifolds of contact manifolds, whose definition recently has been extended to singular Legendrians. We describe a way of constructing simpler models of this dg-algebra for singular Legendrian knots in $\mathbb{R}^{3}$, and give the first examples of singular knots for which the full cohomology can be computed. This includes the $A_{n}$-Legendrians. An important question in the study of the Chekanov-Eliashberg dg-algebra is to understand how its quasi-isomorphism type changes when the Legendrian undergoes Weinstein ribbon isotopy. By explicit computation we show that quite dramatic changes are possible and that among other things, Weinstein ribbon isotopic Legendrians can have Koszul dual dg-algebras. Along the way, we compute the cohomology of the Chekanov-Eliashberg dg-algebra in the boundary of a Weinstein surface. This finishes the proof of the Bourgeois-Ekholm-Eliashberg surgery formula in dimension two, which was missing from the literature.