How efficiently can one untangle a double-twist? Waving is believing! (Q2363605)

There are several visual demonstrations of untangling the double-twist loop. This paper explains the mathematics behind these physical demonstrations. It explains that formally an untangling of the double-twist is a based nullhomotopy in the space of rotations \(SO(3)\) of three-dimensional space. The paper contains a series of aspects related to the mentioned nullhomotopy. The references also give a lot of examples of visual demonstrations of untangling.