Comment by deadfoxygrandpa
5 days ago
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
5 days ago
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.)
But this implies that a simple 1-knot might completely undo itself if you join it to its mirror. Which I assume people have tried, and doesn't work. Likewise with 2's, 3's etc.
It seems intuitively obvious that there is something deeper going on here that makes these two knots work, where (presumably) many others have failed. Or more interestingly to me, maybe there's something special about the technique they use, and it might be possible to use this technique on any/many pairs of knots to reduce the sum of their unknotting numbers.
1 reply →
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.
"counterexample: an example that refutes or disproves a proposition or theory"
Yes, the article is about it ... which has no bearing on my point, and just repeats the logic error.
It is frequently the case that a counterexample is obviously (or readily seen to be) a counterexample to a conjecture. That has no bearing on how long it takes to find the counterexample. e.g., in 1756 Euler conjectured that there are no integers that satisfy a^4+b^4+c^4=d^4 It took 213 years to show that 95800^4+217519^4+414560^4=422481^4
satifies it ... "obviously".
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.
Not quite 'obviously', but mathematical folklore has it that 'clearly' is used to mark the difficult conceptual step in a proof.