Why Mathematicians Study Knots

David S. Richeson in Quanta:

Knot theory began as an attempt to understand the fundamental makeup of the universe. In 1867, when scientists were eagerly trying to figure out what could possibly account for all the different kinds of matter, the Scottish mathematician and physicist Peter Guthrie Tait showed his friend and compatriot Sir William Thomson his device for generating smoke rings. Thomson — later to become Lord Kelvin (namesake of the temperature scale) — was captivated by the rings’ beguiling shapes, their stability and their interactions. His inspiration led him in a surprising direction: Perhaps, he thought, just as the smoke rings were vortices in the air, atoms were knotted vortex rings in the luminiferous ether, an invisible medium through which, physicists believed, light propagated.

Although this Victorian-era idea may now sound ridiculous, it was not a frivolous investigation. This vortex theory had a lot to recommend it: The sheer diversity of knots, each slightly different, seemed to mirror the different properties of the many chemical elements. The stability of vortex rings might also provide the permanence that atoms required.

More here.