to make any THK knot, maybe we should count first, how many THK knots indeed exist?

for a THK(p,b), with p parts and b bights,

the only requirement is the Law of Common Devisor, p and b should be co-prime.

` gcd(p,b) = 1`

any pair of co-prime positive natual numbers can represent a THK knot, THK(5,3) or THK(4,11) etc...

so how many pairs of co-prime positive natual numbers exist?

obviously it is infinite.

technically speaking, as many as "rational number"

by definition, "rational number" means a number that can be expressed as the quotient of an integer divided by a nonzero integer

rational number and THk can have a one-to-one map, so the count of them are equal.

so, although people sometimes like to use a multiply mark x to express a THK, like 3x2, 5x4,

the better way is to use a slash, which usally means devide: THK(3/2), THK(5/4)