And yet it almost always works. There were no known counterexamples where it failed until this was published.

crazygringo 3 days ago | parent | next [-]

Yup. I've had lots of intuitions for things, only to discover there was a very non-intuitive theorem conclusively proving my intuition wrong.

So much of math and physics is discovering these beautiful, surprisingly non-intuitive things.

And this fits right in that pattern -- it seems intuitive that it wouldn't be true, but nobody's been able to find a counterexample. So it's yet another counterintuitive result that math is built on. Not proven, but statistically robust.

Which is what makes it great when somebody does ten years of work in simulating knots so a counterexample can be found.

Which doesn't even confirm the original intuition, because there are still so many cases where the rule holds. Whereas our intuition would have assumed a counterexample would have been easy to find, and it wasn't.

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jlarocco 3 days ago | parent | prev [-]

I'm with the OP on this one. Intuitively (to me, anyway) I wouldn't expect it to work in general.

I'm surprised it took so long to find a counterexample, but it doesn't surprise me at all to hear it doesn't work.

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furyofantares 3 days ago | parent [-]

Mine too, but this is not very interesting, although our intuition was right, it was almost certainly right for the wrong reasons.

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jlev1 3 days ago | parent [-]

Agreed, this is the real takeaway for those who think it’s unsurprising: they simply haven’t understood why it is surprising.

	▲	jlarocco 3 days ago \| parent \| next [-]
		Definitely! My understanding of knot theory is limited to having watched a few YouTube videos and reading the first introductory chapters of a book. A neat topic, but not one I'm going to dig too deeply into.
	▲	nenenejej 3 days ago \| parent \| prev [-]
		If that is the case the counterexample is the sort of thing a stubborn cynical and amateur mathematician (and programmer) may have found.