Maybe I'm really dumb, but it should be obvious that replacing a section of rope in one knot with another, is intuitively not going to simply "add the unknotting numbers"
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.
AlotOfReading|5 months ago
crazygringo|5 months ago
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.
jlarocco|5 months ago
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.