A M\"{o}bius invariant discretization and decomposition of the M\"{o}bius energy

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.