What was the immediate purpose of cataloging knot configurations by Kelvin and Tait?

To test the vortex theory of atoms

The initial intensive effort by Thomson (Kelvin) and Tait to meticulously catalog every possible knot configuration was directly tied to their physical hypothesis concerning atomic structure. They believed the vortex theory of atoms—that elements were stable knots in the aether—was correct. Therefore, the act of building physical knot models and creating the first comprehensive knot tables was undertaken as an empirical method to see if a one-to-one correspondence could be established between every observed chemical element and a specific, unique knot type. Although this physical theory ultimately failed to adequately explain chemistry, the rigorous classification effort it necessitated provided the mathematical scaffolding upon which knot theory, as a pure discipline, was later built.