It wasn't the properties of knots, but rather the hydrodynamical properties of closed fluids. It stems from basic facts in fluid mechanics (so curvatures and gradients and Stokes' theorem stuff). In the late 1860's Kelvin proved (assuming inviscid flows) that a closed curve of fluid particles has its circulation 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 in particular say that vortex tubes are frozen into the fluid. As a corollary, interlinked vortex tubes preserve their topology as they are pushed around. It was this "state of permanence" that led to Kelvin's weird theory.

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.