This is very speculative question and I do not know where to start looking up the literature, or if what I am looking for is even mathematically possible/meaningful.

Q: I am interested in finding out whether any variational or action principles have been derived anywhere in the physics or mathematical literature (e.g. in study of geometry of configuration spaces etc.) on the space of braids (or knots).

Explicitly, some example where a Lagrangian is a function of a braid representation, and it is minimized/extremized under some constraints. The resulting equation would be differential equation whose solution gives a trajectory in the space of braids.