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.