Energy Minimization of Knotted Vortex Tubes
Physical and Biological Sciences
MATH 195 - Senior Thesis
Topology is the study of properties preserved under structural deformation, called invariants. Applications of topology ranging from fluid and statistical mechanics to DNA unpacking in molecular biology all center around knot theory. This paper specifically applies knot-theoretic techniques to the study of local rotation in fluids, termed vortex dynamics, which is imperative for efficient vehicle design, aeronautics, wind- and water-powered energy generation, and a variety of other fields.
When an isolated region of rotational fluid forms knotted structures, topological identification of knot type and associated knot properties allows for the determination of certain fluid properties which, under the appropriate constraints, are preserved by fluid motion. The paper’s primary goal is to prove that energy is trapped within knotted vortex structures and that both the quantity of energy and the tendency of said structures to expand or shrink in length can be determined using topological invariants.
