Knotted Fields (Minipgm)

Coordinators: Mark Dennis, William Irvine, Randall Kamien, Robert Kusner

To tie a shoelace into a knot is a relatively simple affair. Tying a knot in a field is a different story, because the whole of space must be filled in a way that matches the knot being tied at the core. The possibility of such localized “knottedness” in a space-filling field has fascinated physicists and mathematicians ever since Kelvin’s “vortex atom” hypothesis, in which the atoms of the periodic table were hypothesized to correspond to closed vortex loops of different knot types in a perfect Eulerian fluid.

In recent years, the idea of knotted fields has gained a renewed interest, as the role of topology has been appreciated in a range of physical systems, including nonlinear field theories, liquid crystals, quantum wavefunctions, condensates, and optical fields.  Crucially, experimental techniques are at a stage now where knotted and linked topological defect filaments have been engineered in laser light and in liquid crystals.  In these cases, the field is some function of 3D space (complex scalar optical amplitude, or a liquid crystal nematic director, in which the defects are nodal and disclination lines, respectively).  Meanwhile, there has been continued interest in topological fluid dynamics, as well as developments in relevant branches of abstract knot theory, including knotted solutions of PDEs and fibered knot theory.  At the same time, topological braiding has become a cornerstone of quantum computing and addresses similar questions of manipulation and measurement of entangled fields. Collectively, these manifestations create a rich and substantive landscape for the exploration of unifying ideas.

This program will bring together experts in these diverse fields; in particular, physicists working on knotted fields in the laboratory, mathematicians interested in applying tools of topology, and theorists interested in proposing new phases of matter and light.