On some (multi)symplectic aspects of link invariants

In this note we construct a homotopy co-momentum map (a` la Ryvkin, Wurzbacher and Zambon, RWZ) trangressing to the standard hydrodynamical co-momentum map of Arnol’d, Marsden and Weinstein and others, then generalized to a special class of Riemannian manifolds. As a byproduct, a covariant phase space interpretation of Brylinski’s manifold of mildly singular links is exhibited upon resorting to the Euler equation for perfect fluids. A semiclassical interpretation of the HOMFLYPT polynomial is also given, building on the Liu-Ricca hydrodynamical approach to the latter and on the Besana-S. symplectic approach to framing. We finally reinterpret the (Massey) higher order linking numbers in terms of conserved quantities within the RWZ multisymplectic framework and determine knot theoretic analogues of first integrals in involution.