Comment by Antinumeric
5 days ago
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.
5 days ago
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.
Yes this is an interesting case where something that seems obvious on first thought also seems like it would be wrong once you try it out, and then after 100 years of trying someone looks hard enough at their plate of spaghetti and realises it was right all along.
It happens: once you see the example, it may be trivial to understand. The hard thing is to find it.
> This example seems obvious to me
The counterexample has 7 crossings. Try to explain why the equivalent knot with only 5 crossing is not an counterexample and you may realize why it's not obvious.
you're either lying or you don't understand what you're looking at. theres a reason this conjecture wasnt disproven for almost a hundred years
I'm not saying I could have come up with the example. I'm saying looking at the example, and seeing how the two unders are connected togther, and the two overs connected together, makes it obvious that there is more freedom to move the knot around. And that freedom, at least to me, is intuitively connected to the unknotting number.
And that is why the mirror image had to be taken - you need to make sure that when you join it is over to over and under to under.
You’re getting a lot of pushback here, but I have to say, your intuition makes sense to me too.
When you’re connecting those two knots, it seems like you have the option of flipping one before you join them. It does seem very plausible that that extra choice would give you the freedom to potentially reduce the knotting number by 1 in the combined knot.
(Intuitively plausible even if the math is very, very complex and intractable, of course.)
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Please don’t jump straight to “lying”, it’s better to assume good faith. I agree it’s likely much more complex than they’re assuming.
Surely the example can be "obvious" because it's simple/clear. I don't think they're commenting on whether _finding_ the example is obvious...
did you look at the example? it's incredibly complicated
Logic fail. The example is not the conjecture. Saying the example is obvious is not saying that the conjecture is obvious.
The example isn't an example -- it's a proposed simplicity of a counterexample. Which is exactly what the article is about and the post you responded to is therefore objecting to.
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P.S. To clarify:
Saying that the counterexample is a posteriori obvious is not saying that the conjecture is a priori obviously false.
I think this is one of those language barrier things. Non-mathematicians sometimes say ‘obvious’ when what they mean is ‘vaguely plausible’.
A math professor at my uni said that a statement in mathematics is “obvious” if and only if a proof springs directly to mind.
If that is indeed the standard, then it's easy to see how something that is vaguely plausible to an outsider can be obvious to someone fully immersed in the field.
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