Before revision:  It is well-known that for a simple isotropic wormlike chain model, the minimal-energy shape of a loop can be derived exactly and is expressed as a combination of inverse elliptic integrals. Here, we construct a simple explicit two-parameter variational solution, referred to as harmonic loop, which reproduces the exact loop energy up to the fourth significant digit .
After revision:  It is well-known that for a simple isotropic wormlike chain model, the excess twist density oscillates as a response to an imposed bending stress, as recently found in DNA minicircles and observed in nucleosomal DNA. These twist oscillations, here referred to as twist waves, are due to the coupling between twist and bending deformations, which in turn originates from the asymmetry between DNA major and minor grooves. We introduce a simple analytical variational shape, that reproduces the exact loop energy up to the fourth significant digit .