How do we know how many prime knots there are for a specific number of crossings?

Question

For instance, on wikipedia it says there are 7 prime knots with 7 crossings. How do we know there isn't an 8th prime knot knot with 7 crossings that we haven't yet discovered?

Answer 1

It seems the only way is to enumerate over all possible diagrams up to the desired number of crossings and then do lots of hard work to eliminate the non-prime knots and remove all duplicates.

For 7 crossings you can probably do this easily by hand with a little patience. For more crossings a computer is desirable.

In fact, 7 crossings should be straightforward as every knot with 7 or fewer crossings is alternating, and its easy to check when alternating diagrams are the same, since the only isotopies between alternating diagrams you need to consider are flypes.

Thistlewaite, Hoste and Weeks are responsible for compiling knot tables for relatively large crossing numbers (up to 18 crossings now, I think). There is a nice overview of the menagerie of techniques they use here (Warning: ~3mb pdf):

http://pzacad.pitzer.edu/~jhoste/HosteWebPages/downloads/HTW.pdf