# Do there exist any variational principles on the space of braids (or knots)?

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.

• Feb 1, 2020 at 0:27
• Energy functionals are also used for braids. It goes back to Graham Segal's work, but if you are interested a more modern manifestation see Paolo Salvatore's recent paper "A cell decomposition of the Fulton Mac Pherson operad". Feb 1, 2020 at 1:55
• @RyanBudney Thanks. Do you happen to have a reference for Segal's work ? Feb 1, 2020 at 16:54
• Configuration-spaces and iterated loop-spaces. Graeme Segal. Inventiones mathematicae volume 21, pages 213–221(1973) Feb 2, 2020 at 0:03
• My Phd supervisor developed a floer theory on spaces of braids: see few.vu.nl/~vdvorst/NEWPAPERS/IM1f.pdf and later work. Mar 21, 2020 at 18:56

For braids, one can look at various variational principles by looking at metrics on Teichmuller space of the punctured sphere. The Teichmuller metric is a Finsler metric, with unique minimizing geodesics for pseudo-Anosov braids. The Weil-Petersson metric is a Riemannian metric which is incomplete, but for which pseudo-Anosovs also have unique minimizing representatives. The minimizing geodesic then gives a canonical braid representative up to conjugacy (by choosing the representatives to have fixed center of mass and constant moment). One can also find unique representatives for braids up to choosing basepoint surfaces (e.g. $$n$$ points lying on the $$x$$-axis in $$\mathbb{R}^2$$).
One can also look at solutions to the planar $$n$$-body problem to get braid realizations. Richard Montgomery first proved the existence of solutions to the 3-body problem.