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We study a class of nonlocal, energy-driven dynamical models that govern themotion of closed, embedded curves from both an energetic and dynamicalperspective. Our energetic results provide a variety of ways to understandphysically motivated energetic models in terms of more classical, combinatorialmeasures of complexity for embedded curves. This line of investigationculminates in a family of complexity bounds that relate a rather broad class ofmodels to a generalized, or weighted, variant of the crossing number. Ourdynamic results include global well-posedness of the associated partialdifferential equations, regularity of equilibria for these flows as well as amore detailed investigation of dynamics near such equilibria. Finally, weexplore a few global dynamical properties of these models numerically.

Dynamics of Embedded Curves by Doubly-Nonlocal Reaction-Diffusion Systems