  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.
