2+1-dimensional traversable wormholes supported by positive energy

We revisit the shapes of the throats of wormholes, including thin-shell wormholes (TSWs) in $2+1-$dimensions. In particular, in the case of TSWs this is done in a flat $2+1-$dimensional bulk spacetime by using the standard method of cut-and-paste. Upon departing from a pure time-dependent circular shape i.e., $r=a\left( t\right)$ for the throat, we employ a $$dependent closed loop of the form $r=R\left( t,\theta \right) ,$ and in terms of $R\left( t,\theta \right)$ we find the surface energy density $\sigma$ on the throat. For the specific convex shapes we find that the total energy which supports the wormhole is positive and finite. In addition to that we analyze the general wormhole's throat. By considering a specific equation of $r=R\left( \theta \right)$ instead of $r=r_{0}=const.,$ and upon certain choices of functions for $R\left( \theta \right)$ we find the total energy of the wormhole to be positive.