The very first catalogue of different types of knots dates from 1876, and was the brainchild of the Scottish physicist Peter Guthrie Tait. In fact, Tait believed he was constructing the periodic table of the elements: together with his friend and fellow physicist William Thomson (later Lord Kelvin), he had developed a theory of physics in which atoms were knotted vortices in an all-pervading aether. According to that hypothesis, to tell chemical elements apart, you had to tell knots apart.
The theory of vortex-atoms was short-lived. But Tait’s labour of love, classifying different knots according to their number of crossings, continues to this day.
By 1877, Tait had single-handedly got as far as knots with 7 crossings. He was joined in his project by the Reverend Thomas Kirkman, a mathematical vicar from Lancashire in the UK, and Charles Little of the University of Nebraska. Working individually and together, they largely managed to classify knots with 8, 9, and 10 crossings, and made inroads into those with 11. You can read some of their original papers here.
In 1998, Hoste, Thistlethwaite, and Weeks announced a classification up to 16 crossings, amounting to 1,701,936 distinct knots.
Although no complete classification is yet known for knots or links (i.e ‘knots’ which involve more than one piece of string) with more than 16 crossings, important subfamilies have been analysed: namely prime, alternating links.
The work of Flint and Rankin has focused on these families, and their recent results show that…
“In total, there are 98,517,495,461 prime alternating links of crossing size at most 23, and there are 417,377,448,058 prime alternating links of 24 crossings.”