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I read the Quanta article on this when it came out. They show the knots, and they're simple enough that I was almost surprised that the counterexample hadn't been found before. But seeing the shockingly complicated unknotting procedure here makes it much clearer why it wasn't!

It's interesting that you have to first weave the knot around itself, which adds many more crossings. Only then do you get a the special unknotting that falsifies the conjecture.

Whenever I encounter this sort of abstract math (at least “abstract” for me) I start wondering what’s even “real”. Like, what is some foundational truth of reality vs. stuff we just made up and keep exploring.

Are these knots real? Are prime numbers real? Multiplication? Addition? Are natural numbers really “natural”?

For example, one thing that always seemed bizarre to me for as long as I can remember is Pi. If circles are natural and numbers are natural, then why does their relationship seem so unnatural and arbitrary?

You could imagine some advanced alien civilization, maybe in a completely different universe, that isn’t even aware of these concepts. But does it make them any less real?

Sorry for rambling off topic like a meth addict, just hoping someone can enlighten me.

Great video coverage from Stand-up Maths https://www.youtube.com/watch?v=Dx7f-nGohVc

This example seems obvious to me - Joining the under to the under, and the over to the over would obviously give more freedom to the knot than the reverse.