I've often heard that Lord Kelvin was one of the first people to study knot theory, as he hypothesized that atoms were knots in the ether. I assume that he had some compelling evidence for this fact.
What properties of knots did Lord Kelvin consider similar to atomic properties?
It wasn't the properties of knots, but rather the hydrodynamical properties of closed fluids. It stems from the most basic facts in fluid mechanics:
Kelvin proved (assuming inviscid flows) that a closed curve $C$ of fluid particles (velocity field $u$) has its circulation $\oint_Cu\cdot dl$ independent of time. His theorem isn't true if the curve is fixed in space -- it literally must be a material curve that can move with the fluid.
This is closely related to Helmholtz's laws, which say that vortex tubes are frozen into the fluid. The relation: vorticity is by definition $\omega=\nabla\times u$, and Stokes' theorem shows that the flux of vorticity $\int_S\omega\cdot dS$ is precisely the circulation along slices $S$ of the vortex tube.
As a corollary, interlinked vortex tubes preserve their topology when being pushed around. It was this "state of permanence" that led to Kelvin's weird theory.
Alongside the hydrodynamic aspects of the proposal, which included Kelvin and Tait experimenting with smoke rings bouncing off each other, an important aspect of knots was their variety. If atoms were some kind of hard sphere, it seemed hard to explain the very varied properties of the elements, around sixty of which had been discovered by the 1860s. The portion of the task set for Tait was to tabulate the different knot types. These were then to be associated to the different elements, with the hope that vibrations in the knotted vortice types could account for the different emission spectra of elements.
There was even a religious motivation for them both. Some agent was needed to set the vortices in motion. You can read about all this in a brief history of Kelvin and Tait's work I wrote here.
3 seconds of google found me an online copy of Kelvin's paper. The short answer is that he wasn't just hypothesizing about knots, but rather knotted vortices. Interactions between vortices and "elastic" vibrations of vortices are two points that he mentions.
It wasn't just Kelvin---the three premier Scottish physicists of the 19 th century, Maxwell, Thomson (Kelvin) and Tait were all fascinated by knots. Some details can be found in the article by D.S. Silver "Scottish Physics and knot theory's odd origins" which is available online. The fact that this didn't lead to important advances in physics should not blind one to their other seminal contributions and it is arguable that this work led to the development of modern knot theory.
