method to tie any Turks Head Knots (THK)

General Discussion
1

Every few years, someone would come forward and claim that he had found a new way to make arbitrary Turks Head Knots(THK).

But in fact, at least as early as about 30 years ago(about year 1995 ), experts such as TomHall Schaake have already studied this thoroughly and published books to explain methods for making all THK.

what a myth

2

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)

3

You only mention the string run options for O1U1. If you take all possible patterns into account there will be quite a few more options, not?

4

actually for THK , the O1U1 pattern does not matter at all.

the only requirement is co-prime. so that you can return to the begin point and finish it using single line.

pattern is another thing , O1U1 OR O2U2 , or any irregular pattern.

but ususally only think of O1U1 pattern.