Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

The Unknotting Number Is Not Additive

divisbyzero.com

Post reply on HN
Print view View original

41–50 of 80 posts

Re: The Unknotting Number Is Not Additive

#41

Post by gcanyon »

iainmerrick wrote: post #28
Antinumeric wrote: post #25

Earlier quoted context omitted.

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…

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.

Re: The Unknotting Number Is Not Additive

#42

Post by gcanyon »

CJefferson wrote: post #11
brap wrote: post #3

Whenever 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…

To me, the least real thing in maths is, ironically, the real numbers. As you dig through integers, fractions, square roots, solutions to polynomials, things a turing machine can output, you get to increasingly large classes of numbers which are still all countably infinite. At some point I realised I'd covered anything I could ever imagine caring about and was still in a countable set.

You might appreciate this video where Matt Parker lays out the various classes of numbers and concludes by describing the normal numbers as being the overwhelmingly vast proportion of numbers and laments "we mathematicians think we know what's what, but so far we have found none of the numbers."

https://www.youtube.com/watch?v=5TkIe60y2GI

Re: The Unknotting Number Is Not Additive

#43

Post by James_K »

Antinumeric wrote: post #21

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.

Re: The Unknotting Number Is Not Additive

#44

Post by amiga386 »

brap wrote: post #3

Whenever 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'm fairly confident that most mathematics are real, i.e. they have real world analogues. Pi is just an increasingly close look at the ratio between a circle's diameter and circumference.

I'm willing to believe elecromagnetic fields are real - you can see the effects magnets (and electromagnets) have on ferrous material. You can really broadcast electromagnetic waves, induce currents in metals, all that. I'm willing to believe atoms, quarks, electrons, photons, etc. are real. Forces (electrical charge, weak and strong nuclear force, gravity) are real.

What I'm not willing to believe is that quantum fields in general are real, that physical components are not real and don't literally move, they're just "interactions" with and "fluctuations" in the different quantum fields. I refuse to believe that matter doesn't exist and it's merely numbers or vectors arranged a grid. That's a step too far. That's surely just a mathematical abstraction. And yet, the numbers these abstractions produce match so well with physical observations. What's going on?

Re: The Unknotting Number Is Not Additive

#45

Post by eprparadox »

brap wrote: post #3

Whenever 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…

there's a great episode of Mindscape where Max Tegmark takes this idea and runs with it: https://www.preposterousuniverse.com/podcast/2019/12/02/75-m...

Re: The Unknotting Number Is Not Additive

#46

Post by jibal »

gcanyon wrote: post #38
jibal wrote: post #32

Earlier quoted context omitted.

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".

Re: The Unknotting Number Is Not Additive

#47

Post by kannanvijayan »

schiffern wrote: post #40
kannanvijayan wrote: post #30

Earlier quoted context omitted.

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 an…

>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. Easily fixed! I choose 1 dimension. :)

Hah, nice find :)

Re: The Unknotting Number Is Not Additive

#48

Post by BobbyTables2 »

amiga386 wrote: post #44
brap wrote: post #3

Whenever 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'm fairly confident that most mathematics are real, i.e. they have real world analogues. Pi is just an increasingly close look at the ratio between a circle's diameter and circumference. I'm willing to believe elecromagnetic fields are real - you can see the effects magnets (and electromagnets) have on ferrous material. You can really broadcast electromagnetic waves, induce currents in metals, all that. I'm willing…

What about the particles that randomly pop in and out of existence?

If one thinks about it, electromagnetism is really bizarre.

How can two electrons actually repel each other? Sure, they do, but it’s practically witchcraft.

Magnetism is even more weird.

Re: The Unknotting Number Is Not Additive

#49

Post by amiga386 »

BobbyTables2 wrote: post #48
amiga386 wrote: post #44

Earlier quoted context omitted.

I'm fairly confident that most mathematics are real, i.e. they have real world analogues. Pi is just an increasingly close look at the ratio between a circle's diameter and circumference. I'm willing to believe elecromagnetic fields are real - you can see the effects magnets (and electromagnets) have on ferrous material. You can really broadcast electromagnetic waves, induce currents in metals, all that. I'm willing…

What about the particles that randomly pop in and out of existence? If one thinks about it, electromagnetism is really bizarre. How can two electrons actually repel each other? Sure, they do, but it’s practically witchcraft. Magnetism is even more weird.

> What about the particles that randomly pop in and out of existence?

I like to imagine they're somehow just an observational error, otherwise the https://en.wikipedia.org/wiki/One-electron_universe is real and we get a universe-sized '—All You Zombies—'

> How can two electrons actually repel each other

Indeed. I think it's something we can only intuit, I don't think we've really gotten to the bottom of it. Trying to push two electrons together feels like trying to push a car up a hill, or pressing on springs. The force you fight against is just there and you feel its resistance

Re: The Unknotting Number Is Not Additive

#50

Post by IAmBroom »

brap wrote: post #39
twiceaday wrote: post #20

Earlier quoted context omitted.

Math is a purely logical tool. None of it "exists." That makes no sense. Some of it can be used to model reality. We call such math "physics." And I think physics is significantly closer to math than to reality. It's just a collection of math that models some measurements on some scales with some precision. We have no idea how close we are to actual reality. I do not understand the framing of "translating math concep…

I think maybe I didn’t really explain myself properly. I didn’t mean that math is real in the sense that atoms are real. Perhaps “true” would be a better word. We know these things are true to us, but are they universally true? If that’s even a thing? Hope that makes more sense.

The age-old problem of a respondent using different definitions of words than the OP.

Socrates made a whole career out of it.

Post reply on HN