We prove that any regular integral invariant of volume-preserving transformations is
equivalent to the helicity. Specifically, given a functional ℐ defined on exact divergence-free …

Geometric Hydrodynamics has flourished ever since the celebrated 1966 paper of V. Arnold.
In this paper we present a collection of open problems along with several new constructions …

Many invariants of knots and links have their counterparts in braid theory. Often, these
invariants are most easily calculated using braids. A braid is a set of n strings stretching …

This paper employs higher-order winding numbers to generate Hamiltonian motion of
particles in two dimensions. The ordinary winding number counts how many times two …

Three-component links in the 3-dimensional sphere were classified up to link homotopy by
John Milnor in his senior thesis, published in 1954. A complete set of invariants is given by …

We consider the general non-vanishing, divergence-free vector fields defined on a domain
in-space and tangent to its boundary. Based on the theory of finite-type invariants, we define …

We introduce an alternative approach to the third order helicity of a volume preserving vector
field B, which leads us to a lower bound for the L 2-energy of B. The proposed approach …

To each three-component link in the 3-sphere we associate a generalized Gauss map from
the 3-torus to the 2-sphere, and show that the pairwise linking numbers and Milnor triple …

We revisit an interpretation of higher-dimensional helicities and Hopf–Novikov invariants
from the point of view of the Brownian ergodic theorem. We also survey various results …

If the vorticity field of an ideal fluid is tangent to a foliation, additional conservation laws
arise. For a class of zero-helicity vorticity fields, the Godbillon-Vey (GV) invariant of foliations …