Structural complexity of vortex flows by diagram analysis and knot polynomials

In this paper I present and discuss with examples new techniques based on the use of geometric and topological information to quantify dynamical information and determine new relationships between structural complexity and dynamical properties of vortex flows. New means to determine linear and angular momenta from standard diagram analysis of vortex tangles are provided, and the Jones polynomial, derived from the skein relations of knot theory is introduced as a new knot invariant of topological fluid mechanics. For illustration several explicit computations are carried out for elementary vortex configurations. These new techniques are discussed in the context of ideal fluid flows, but they can be equally applied in the case of dissipative systems, where vortex topology is no longer conserved. In this case, a direct implementation of adaptive methods in a real-time diagnostics of real vortex dynamics may offer a new, powerful tool to analyze energy-complexity relations and estimate energy transfers in highly turbulent flows. These methods have general validity, and they can be used in many systems that display a similar degree of self-organization and adaptivity.