  Recently, Duminil-Copin and Smirnov proved a long-standing conjecture by
Nienhuis that the connective constant of self-avoiding walks on the honeycomb
lattice is $\sqrt{2+\sqrt{2}}.$ A key identity used in that proof depends on
the existence of a parafermionic observable for self-avoiding walks on the
honeycomb lattice. Despite the absence of a corresponding observable for SAW on
the square and triangular lattices, we show that in the limit of large
lattices, some of the consequences observed on the honeycomb lattice persist on
other lattices. This permits the accurate estimation, though not an exact
evaluation, of certain critical amplitudes, as well as critical points, for
these lattices. For the honeycomb lattice an exact amplitude for loops is
proved.
