  We investigate a discrete version of the M\"obius energy, that is of
geometric interest in its own right and is defined on equilateral polygons with
$n$ segments. We show that the $\Gamma$-limit regarding $L^{q}$ or $W^{1,q}$
convergence, $q\in [1,\infty]$ of these energies as $n\to\infty$ is the smooth
M\"obius energy. This result directly implies the convergence of almost
minimizers of the discrete energies to minimizers of the smooth energy if we
can guarantee that the limit of the discrete curves belongs to the same knot
class. Additionally, we show that the unique minimizer amongst all polygons is
the regular $n$-gon. Moreover, discrete overall minimizers converge to the
round circle.
