  We study a class of nonlocal, energy-driven dynamical models that govern the
motion of closed, embedded curves from both an energetic and dynamical
perspective. Our energetic results provide a variety of ways to understand
physically motivated energetic models in terms of more classical, combinatorial
measures of complexity for embedded curves. This line of investigation
culminates in a family of complexity bounds that relate a rather broad class of
models to a generalized, or weighted, variant of the crossing number. Our
dynamic results include global well-posedness of the associated partial
differential equations, regularity of equilibria for these flows as well as a
more detailed investigation of dynamics near such equilibria. Finally, we
explore a few global dynamical properties of these models numerically.
