  The M\"{o}bius energy, defined by O'Hara, is one of the knot energies, and
named after the M\"{o}bius invariant property which was shown by
Freedman-He-Wang. The energy can be decomposed into three parts, each of which
is M\"{o}bius invariant, proved by Ishizeki-Nagasawa. Several discrete versions
of M\"{o}bius energy, that is, corresponding energies for polygons, are known,
and it showed that they converge to the continuum version as the number of
vertices to infinity. However already-known discrete energies lost the property
of M\"{o}bius invariance, nor the M\"{o}bius invariant decomposition. Here a
new discretization of the M\"{o}bius energy is proposed. It has the M\"{o}bius
invariant property, and can be decomposed into the M\"{o}bius invariant
components which converge to the original components of decomposition in the
continuum limit. Though the decomposed energies are M\"{o}bius invariant, their
densities are not. As a by-product, it is shown that the decomposed energies
have alternative representation with the M\"{o}bius invariant densities.
