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?



What properties of knots lead Lord Kelvin to hypothesize that atoms were knots in the ether?
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.


