As part of the workshop Computer Science for Knotty Math Problems, The School of Theoretical Physics, DIAS presents an Honourary Public Lecture.

Abstract: The theory of knots has its origins in the deep history of weaving and the manifold uses of practical knot tying that culminated in 17th century maritime sailing. Knots have been associated with numbers via the Mayan Quipu methods of bookkeeping that represented number by countable twists in weaves. An extraordinary new view of knotting occurred in the 19th century with the theory of vortex atoms of Lord Kelvin (Sir William Thompson) a leading physicist of that era. Thompson proposed that atoms of matter (just beginning to be recognized) could be identified with knotted vortices in the luminiferous aether.

This vortex theory of atoms depended on the idea, nay the very existence, of an all-pervasive fluid that filled all of space and whose undulations would transmit the electromagnetic field and whose swirlings would provide the energy and form for the matter of the universe, its atoms. Alas, this theory did not last. The luminiferous aether was replaced by the abstractions of differential geometry and more lately networks and pre-geometry, strings, higher dimensions. Atoms and knots may be related but not so directly.

Nevertheless, the idea of knotted vortices continues to excite the imaginations of mathematicians and physicists. It was not until 2012 that scientists actually saw a knotted vortex in a fluid. The medium was water. The experimenters were Dustin Kleckner and William Irvine at the University of Chicago. Topological vortices such as these are short lived, undergoing reconnection of their arcs and degenerating into unknotted collections of circles.

In this talk we will show how one can use knot theory to study the reconnection properties of vortices and how recent topological work such as Khovanov homology applies to give lower bounds on the number of reconnections needed to undo a vortex. We will show movies of real experiments and computer models of vortices and of reconnection of vortices in the work of Kleckner and Irvine and in the work of Aleeksenko. All of this leads to a need to reassess the work of Kelvin. Time permitting, we will discuss problems of unknotting knots via the use of self-repelling fields of charge along the knots. These specific examples illustrate the interactions between topological theory, the physics of actual knotting, the speculative aspects of models in the physical realm and how computing and topological physics are intertwined.