  Topological Josephson junctions (TJJs) have been a subject of widespread
interest due to their hosting of Majorana zero modes. In long junctions, i.e.
junctions where the junction length exceeds the superconducting coherence
length, TJJs manifest themselves in specific features of the critical current.
Here we propose to couple the helical edge states mediating the TJJ to
additional channels or quantum dots, by which the effective junction length can
be increased by tunable parameters associated with these couplings, so that
such measurements become possible even in short junctions. Besides effective
low-energy models that we treat analytically, we investigate realizations by a
Kane-Mele model with edge passivation and treat them numerically via tight
binding models. In each case, we explicitly calculate the critical current
using the Andreev bound state spectrum and show that it differs in effective
long junctions in the cases of strong and weak parity changing perturbations
(quasiparticle poisoning).
