The fastest sequence of untangling the ropes that was found was this 114-command beauty:

RSSSSRSSRSSRSSRSSRSSRSSRSSRSSRSSRSSRSS
RSSRSSRSSRSSSSSRSSRSSRSSSRSSRSSSSRSSRS
SRSSSSRSSRSSRSSRSSRSSRSSSRSSSRSSRSSRSS

It could be discovered by using the theory of rational tangles, which maps ropes tangled by these commands onto rational numbers using continued fractions. The late John H. Conway of puzzle math fame worked on this, hence our silly name for the dance “Yawn-ock” (“Conway” is hidden in there backwards).

Here are the solvers of this tangly challenge, with the first group having the shortest solutions and the second group being ordered chronologically.