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.
The Unknotting Number Is Not Additive
21–30 of 80 posts
Re: The Unknotting Number Is Not Additive
#22This 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.
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
Re: The Unknotting Number Is Not Additive
#23Whenever I encounter this sort of abstract math (at least “abstract” for me) I start wondering what’s even “real”. Like, what is some foundational truth of reality vs. stuff we just made up and keep exploring. Are these knots real? Are prime numbers real? Multiplication? Addition? Are natural numbers really “natural”? For example, one thing that always seemed bizarre to me for as long as I can remember is Pi. If circ…
Theres always an xkcd : https://xkcd.com/435/
Nice line, but it isn't fully complete. After the Mathematicians there's Logic - and Philosophy - And in the end you complete the circle and go all back to Sociology again.
One issue I sometimes witnessed myself was that Mathematicians sometimes form Groups that behave like pathological examples from Sociology. E.g. there was the Monty-Hall problem, where societies of mathematicians had a meltdown. Sadly I've seen this a few times when Sociology/Mass psychology simply trumped Math in Power.
Re: The Unknotting Number Is Not Additive
#24This 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.
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
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...
Re: The Unknotting Number Is Not Additive
#25This 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.
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.
Re: The Unknotting Number Is Not Additive
#26This 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.
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 think this is one of those language barrier things. Non-mathematicians sometimes say ‘obvious’ when what they mean is ‘vaguely plausible’.
Re: The Unknotting Number Is Not Additive
#27This 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.
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
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.
Re: The Unknotting Number Is Not Additive
#28Earlier quoted context omitted.
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…
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.)
Re: The Unknotting Number Is Not Additive
#29Earlier quoted context omitted.
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 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.
Re: The Unknotting Number Is Not Additive
#30Whenever I encounter this sort of abstract math (at least “abstract” for me) I start wondering what’s even “real”. Like, what is some foundational truth of reality vs. stuff we just made up and keep exploring. Are these knots real? Are prime numbers real? Multiplication? Addition? Are natural numbers really “natural”? For example, one thing that always seemed bizarre to me for as long as I can remember is Pi. If circ…
I don't have an answer to your questions, but I think these thoughts are not uncommon for people who get into these topics. The relationship between the reals, including Pi, and the countables such as the naturals/integers/rationals is suggestive of some deeper truth.
The ratio between the areas of a unit circle (or hypersphere in whatever dimension you choose) and a unit square (or hypercube in that dimension) in any system will always require infinite precision to describe.
Make the areas between the circle and the square equal, and the infinite precision moves into the ratio between their lower order dimensional measures (circumfence, surface area, etc.).
You can't describe a system that expresses the one, in terms of a system that expresses the other, without requiring infinite precision (and thus infinite information).
Furthermore, it really seems like a bunch of the really fundamental reals (pi, e), have a pretty deep connection to algebras of rotations (both pi and e relate strongly to rotations)
What that seems to suggest to me is that if the universe is discrete, then the discreteness must be biased towards one of these modes or the other - i.e. it is natively one and approximates the other. You can have a discrete universe where you have natural rotational relationships, or natural linear relationships, but not both at the same time.