In this article a homotopy co-momentum map (\`a la
Callies-Fr\'egier-Rogers-Zambon) trangressing to the standard hydrodynamical
co-momentum map of Arnol'd, Marsden and Weinstein and others is constructed and
then generalized to a special class of Riemannian manifolds. Also, a covariant
phase space interpretation of the coadjoint orbits associated to the Euler
evolution for perfect fluids and in particular of Brylinski's manifold of
smooth oriented knots is discussed. As an application of the above homotopy
co-momentum map, a reinterpretation of the (Massey) higher order linking
numbers in terms of conserved quantities within the multisymplectic framework
is provided and knot theoretic analogues of first integrals in involution are
determined.Comment: 21 pages, 3 figures. The present version focuses on the connections
between multisymplectic geometry, hydrodynamics and vortices. The derivation
of the HOMFLYPT polynomial via geometric quantization has been proposed as a
separate preprint, see "Derivation of the HOMFLYPT knot polynomial via
helicity and geometric quantization ", arXiv:1910.xxx