curvature - What parameters could be considered to measure the coiling of a ring?

I want to draw a phase diagram to quantify the coiling of a ring inside my system and I was wondering what parameters could be use to precisely define how coiled the ring is.

Currently I am modeling the ring as a regular curve and I am using the torsion and the curvature of the curve as the parameters. The ring exerts a force against the medium that contains it and I was considering this as a possible parameter because it seems to be proportional to the degree of coiling.

What are usual choices to measure the degree of coiling of a ring (inside a viscous medium like the inside of a cell)? Are there any references in the literature that I can be pointed to?

Answer

The mathematical description of a plasmid could help. This is a circular DNA, i.e. a double helical polymer closed on itself. Being double stranded, the polymer can be coiled, and the torsional stress "organizes" the global structure of the ring. Two numbers describe it: twist $Tw$ (turns of the helix around the axis) and writhe $Wr$ (turns of the axis around itself), their sum being the linking number $Lk=Tw+Wr$ (in the usual horrible two letter notation...). Different conformations of the ring with equal linking number are topologically equivalent. See DNA supercoiling for more details.

(The linking number of the DNA is tightly controlled by specialized enzymes which continuously coil and uncoil the double helix to keep an average slightly unwound DNA, which is then more accessible to the other cellular machines)