  The paper describes the unique geometric properties of ideal
magnetohydrodynamics (MHD), and demonstrates how such features are inherited by
extended MHD, viz. models that incorporate two-fluid effects (the Hall term and
electron inertia). The generalized helicities, and other geometric expressions
for these models are presented in a topological context, emphasizing their
universal facets. Some of the results presented include: the generalized Kelvin
circulation theorems; the existence of two Lie-dragged 2-forms; and two
concomitant helicities that can be studied via the Jones polynomial, which is
widely utilized in Chern-Simons theory. The ensuing commonality is traced to
the existence of an underlying Hamiltonian structure for all the extended MHD
models, exemplified by the presence of a unique noncanonical Poisson bracket,
and its associated energy.
