So... the math of knots is... knot theory, and you're correct that it is topology.
How to actually tie knots -- in a practical sense -- would be knotcraft [1] -- not sure what kind of math that would entail? What kind of connection did have in mind between that and the math of knots and programming? (in your original statement)
If it's a mathematical treatment of the strength of knots (physics), I believe that's what the article is about, which is quite interesting.
Well, since this theory does not actually exist yet, and I only have a 'feeling' that it should, I'll have to speculate very hard here:
Something where I have some sort of representation of the physical knot that is much less than just a drawing of the knot, but more than just saying 'this is a reef knot'. Something you can do operations on.
Say you are trying to tie a reef knot. You can tie a reef knot many ways. Say you want to write down how to tie it one way.
So you take 2 strings. Say you represent that by the number 2.
Then you have to twist them together, so you write down something like 2T.
Then you have to pull a bite on each of the 2 strings string. So you write down 2T + 2B.
Then you have to retwist them, but the opposite way. So you write down something like 2T + 2B + 2(-T). Then you have a reef knot.
But the thing is that since there are a LOT of ways to tie a reef knot, that 'formula' should be able to be manipulated to produce the other 'formulas' that also produce a reef knot. Also, a reminder, this 'formula' thingy I just made up should not be taken as any sort of gospel.
We all know that when you pull hard on some knots, that they just decompose into a simpler knot. So this representation of knots should be able to be worked though on paper and give the right result. Or with slip knots and hitches, knots that are designed to decompose into just a string when you pull hard on them or something. So when you take the knot tying 'formula' and then work through it, it just returns a string.
I think you could then apply the physics to these 'formulae', like, each twist you do reduces the rope strength by X% and increased friction by Y%.
Some sort of thing like that is what I am thinking, I guess.
