I sort of remember being in the same position as you a few months ago been throwing for about 3 years and getting really frustrated that I would get knots for 'no apparent reason. I don't want to dismiss you, but it is a lot of experimentation. Try to be cognizant while you're throwing, such as when you make a mistake. Drop one string at a time instead of all of them and see what happens. You'll slowly start to see patterns and common mounts emerge and how you might avoid knots or even use them to your advantage to create more interesting mounts or tricks.
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I could also quote videos, but I would say you should watch videos that you really enjoy and that inspire you. Try to copy what other players are doing, chances are, you're not going to always be able to do it exact and you will end up developing your own style , but this will add to your repertoire of tricks and mounts.
So, continue to practice, try to notice what you are doing I like doing relaxing sessions where I just throw and not worry about labbing trick creation or being very clean , discuss your ideas and struggles with others like you're doing now, and you will be well on your way to understanding knots and more.
I think what you're describing is 1. Monorail is a easy way to explore red triangles. Really, understanding all the crazy twisting 'knotted' tricks comes down to intuition and experience. As you learn, perform and importantly mess up those tricks, you start to understand how the twists and holes and loops all interact with each other. Also, there's a option 3 you missed, the magic knot.
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It is a literal knot a slip knot to be exact in the middle of the string. One of my math roommates in college was a knot theorist in a summer research program. I kept decent track of his research over the summer to understand knot theory enough to know what is needed the most in yoyoing. Honestly I agree that going too deep into knot theory will be more of a distraction than an application to yoyoing and eventually toying with the string and a dead yoyo in the car during a road trip will give better intuition of yoyo knots.
Plenty of the premises of Mathematical Knot Theory are fundamentally different to yoyoing.
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For example in math they study loops of strings called Knots or multiple loops interlinked called Links, but none of the main styles of yoyoing line up with that property easily. If you did crack open a knot theory textbook, really the only important things that could apply to yoyoing is knowing how knot diagrams work, which are very instinctual, a decent basic understanding of the Reidemeister Moves , if they are not already intuitive for you anyway, and understanding a little bit of isotopy can help a little.
All of this above are really just definitions, conventions and axioms of the subject that Mathematicians set before going into a deep study and going deeper than that will get abstract and impractical to yoyoing real quick, like studying Knot Polynomials, its Category Theory, and Knot Invariants is hardly going to help in yoyoing practically. The next natural idea is coming up with a yoyo field with its own knot theory that would work in our system and culture.
After all Juggling gets to have Siteswaps and even Poi has its own theories in tech. Surprisingly there was an early attempt at that in the Yonomicon book written by Mark McBride, given yoyoing back then hardly had any kink mounts and it wasn't addressed, it is definitely obsolete or underdeveloped now. Personally I have thought about it a little bit but its hard to tell if something would catch on or fizzle out like Mark's system from the late 90s. Truthfully, I think you are giving it too much thought. I would say just keep playing and working on things and you will start to understand when you are in a position for a knot.
False knots are something that are good to know how to get into of you're into that kind of thing. Avoiding real knots is all about knowing the elements you are using.
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