7 ms·

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 unknottin

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"

AlotOfReading 1y ago
And yet it almost always works. There were no known counterexamples where it failed until this was published.
- jlarocco 1y ago
  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.
  - furyofantares 1y ago
    Mine too, but this is not very interesting, although our intuition was right, it was almost certainly right for the wrong reasons.
    - jlev1 1y ago
      Agreed, this is the real takeaway for those who think it’s unsurprising: they simply haven’t understood why it is surprising.
      - nenenejej 1y ago
        If that is the case the counterexample is the sort of thing a stubborn cynical and amateur mathematician (and programmer) may have found.
      - jlarocco 1y ago
        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.
- crazygringo 1y ago
  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.

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"

AlotOfReading 1y ago

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

jlarocco 1y ago
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.
- furyofantares 1y ago
  Mine too, but this is not very interesting, although our intuition was right, it was almost certainly right for the wrong reasons.
  - jlev1 1y ago
    Agreed, this is the real takeaway for those who think it’s unsurprising: they simply haven’t understood why it is surprising.
    - nenenejej 1y ago
      If that is the case the counterexample is the sort of thing a stubborn cynical and amateur mathematician (and programmer) may have found.
    - jlarocco 1y ago
      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.
crazygringo 1y ago
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.