[HN Gopher] The Unknotting Number Is Not Additive
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The Unknotting Number Is Not Additive
Author : JohnHammersley
Score : 158 points
Date : 2025-10-09 06:39 UTC (16 hours ago)
(HTM) web link (divisbyzero.com)
(TXT) w3m dump (divisbyzero.com)
| ZiiS wrote:
| Great video coverage from Stand-up Maths
| https://www.youtube.com/watch?v=Dx7f-nGohVc
| PixyMisa wrote:
| Can't recommend this video highly enough. Matt clearly
| demonstrates with physical examples both the question and the
| answer, and why it took so long to find that answer.
| brap wrote:
| 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 circles are natural and numbers
| are natural, then why does their relationship seem so unnatural
| and arbitrary?
|
| You could imagine some advanced alien civilization, maybe in a
| completely different universe, that isn't even aware of these
| concepts. But does it make them any less real?
|
| Sorry for rambling off topic like a meth addict, just hoping
| someone can enlighten me.
| slickytail wrote:
| In the words of Kronecker: "God created the integers, all else
| is the work of man."
| srean wrote:
| Had I been god I would have created _scaled turns_ and left
| the rest for humans.
| fedeb95 wrote:
| I sometimes think about the same things. As of now, my best bet
| is that math is one of the disciplines studying exactly these
| questions.
| fjfaase wrote:
| What is real? There are strong indications that what we
| experience as reality is an ilusion generated by what is
| usually refered to as the subconscious.
|
| One could argue that knots are more real than numbers. It is
| hard to find two equal looking apples and talk about two
| apples, because it requires the abstraction that the apples are
| equal, while it is obvious that they are not. While, I guess,
| we all have had the experience of strugling with untying knots
| in strings.
| jrowen wrote:
| It's more than strong indications. What any individual life
| form perceives is a unique subset or projection of reality.
| To the extent that "one true reality" exists, we are each
| viewing part of it through a different window.
| lqet wrote:
| Philosophical problems regarding the fundamental nature of
| reality aside, this short clip is relevant to your question:
|
| > https://www.youtube.com/watch?v=tCUK2zRTcOc
|
| Translated transcript: Physics is a "Real
| Science". It deals with reality. Math is a structural science.
| It deals with the structure of thinking. These structures do
| not have to exist. They can exist, but they don't have to.
| That's a fundamental difference. The translation of
| mathematical concepts to reality is highly critical, I would
| say. You cannot just translate it directly, because this leads
| to such strange questions like "what would happen if we take
| the law of gravitation by old Newton and let r^2 go to zero?".
| Well, you can't! Because Heisenberg is standing down there.
| twiceaday wrote:
| 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 concepts
| directly into reality." It's backwards. You must have first
| chosen some math to model reality. If you get "bad" numbers
| it has nothing to do with translating math to reality. It has
| to do with how you translated reality into math.
| brap wrote:
| 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.
| IAmBroom wrote:
| The age-old problem of a respondent using different
| definitions of words than the OP.
|
| Socrates made a whole career out of it.
| yujzgzc wrote:
| Yes these knots are real and can be experienced with a simple
| piece of rope.
|
| The prime property of numbers is also very real, a number N is
| prime if and only if arranging N items on a rectangular,
| regular grid can only be done if one of the sides of the
| rectangle is 1. Multiplication and addition are even more
| simply realized.
|
| The infinity of natural numbers is not as real, if what we mean
| by that is that we can directly experience it. It's a useful
| abstraction but there is, according to that abstraction, an
| infinity of "natural" numbers that mankind will not be able to
| ever write down, either as a number or as a formula. So
| infinity will always escape our immediate perception and remain
| fundamentally an abstraction.
|
| Real numbers are some of the least real of the numbers we deal
| with in math. They turn out to be a very useful abstraction but
| we can only observe things that approximate them. A physical
| circle isn't exactly pi times its diameter up to infinity
| decimals, if only because there is a limit to the precision of
| our measurements.
|
| To me the relationship between pi and numbers is not so
| unnatural but I have to look at a broader set of abstractions
| to make more sense of it, adding exponentials and complex
| numbers - in my opinion the fact that e^i.pi = 1 is a profound
| relationship between pi and natural numbers.
|
| But abstractions get changed all the time. Math as an academic
| discipline hasn't been around for more than 10,000 years and in
| that course of time abstractions have changed. It's very likely
| that the concept of infinity wouldn't have made sense to anyone
| 5,000 years ago when numbers were primarily used for
| accounting. Even 500 years ago the concept of a number that is
| a square root of -1 wouldn't have made sense. Forget aliens
| from another planet, I'm pretty sure we wouldn't be able to
| comprehend 100th century math if somehow a textbook time-
| traveled to us.
| jaffa2 wrote:
| Theres always an xkcd : https://xkcd.com/435/
| adornKey wrote:
| 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.
| CJefferson wrote:
| 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.
| JdeBP wrote:
| The entirely opposite perspective is quite interesting:
|
| The "natural numbers" are the biggest mis-nomer in
| mathematics. They are the most _un-Natural_ ones. The numbers
| that occur in Nature are almost always complex, and are
| neither integers nor rationals (nor even algebraics).
|
| When you approach reality through the lens of mathematics
| that concentrates the most upon these countable sets, you
| very often end up with infinite series in order to express
| physical reality, from Feynman sums to Taylor expansions.
| rini17 wrote:
| But you can't really have chemistry without working with
| natural numbers of atoms, measured in moles. Recently they
| decided to explicitly fix a mole (Avogadro's constant) to
| be exactly 6.02214076x10^23 which is a natural number.
|
| Semiconductor manufacturing on nanometer scales deals with
| individual atoms and electrons too. Yes, modeling their
| behavior needs complex numbers, but their amounts are
| natural numbers.
| srean wrote:
| I agree. Had humanity made _turning_ the more fundamental
| operation than _counting_ that would have sped up our
| mathematical journey. The Naturals would have fallen off
| from it as an exercise of counting turns.
|
| The calculus of scaled rotation is so beautiful. The
| sacrificial lamb is the unique ordering relation.
| empath75 wrote:
| how large is the set of all possible subsets of the natural
| numbers?
|
| edit: Just to clarify -- this is a pretty obvious question to
| ask about natural numbers, it's no more obviously
| artificially constructed than any other infinite set. It
| seems to be that it would be hard to justify accepting the
| set of natural numbers and not accepting the power set of the
| natural numbers.
| drdeca wrote:
| Some people (not me) would consider only countably many of
| those subsets to be "possible".
| luc4 wrote:
| One could argue that infinite subsets of the natural
| numbers are not really interesting unless one can
| succinctly describe which elements are contained in them.
| And of course there is only a countable number of such
| sets.
| gcanyon wrote:
| 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
| wat10000 wrote:
| You can encompass them all by talking about numbers that can
| be described. Since you can trivially enumerate all possible
| descriptions, this is countably infinite. By definition, it
| is impossible to describe a number outside that set.
| Byamarro wrote:
| Math is about creating mental models.
|
| Sometimes we want to model something in real life and try to
| use math for this - this is physics.
|
| But even then, the model is not real, it's a model (not even a
| 1:1 one on top of that). It usually tries to capture some
| cherry picked traits of reality i.e. when will a planet be in
| 60 days ignoring all its "atoms"[1]. That's because we want to
| have some predictive power and we can't simulate whole reality.
| Wolfram calls these selective traits that can be calculated
| without calculating everything else "pockets of reducability".
| Do they exist? Imho no, planets don't fundamentally exist,
| they're mental constructs we've created for a group of
| particles so that our brains won't explode. If planets don't
| exist, so do their position etc.
|
| The things about models is that they're usually simplifications
| of the thing they model, with only the parts of it that
| interest us.
|
| Modeling is so natural for us that we often fail to realize
| that we're projecting. We're projecting content of our minds
| onto reality and then we start to ask questions out of
| confusion such as "does my mind concept exist". Your mind
| concept is a neutral pattern in your mind, that's it.
|
| [1] atoms are mental concepts as well ofc
| movpasd wrote:
| I believe this is called epistemic pragmatism in philosophy:
| https://en.wikipedia.org/wiki/Pragmatism
| JdeBP wrote:
| More usually, people imagine the reverse of the advanced alien
| civilizations: that the thing that we and they are most likely
| to have in common is the concept of obtaining the ratio between
| a circle's circumference and its diameter, whereas the things
| that they possibly aren't even aware of are going to be
| concepts like economics or poetry.
| kurlberg wrote:
| Fun historical fact: knot theory got a big boost when lord
| Kelvin (yeah, that one) proposed understanding atoms by
| thinking of them as "knotted vortices in the ether".
| rini17 wrote:
| I see it like natural sciences strive to do replicable
| experiments in outside world, while math strives to do
| replicable experiments in mind. Not everything is transferable
| from one domain to the other but we keep finding many parallels
| between these two, which is surprising. But that's all we have,
| no foundational truths, no clear natural/unnatural divide here.
| kannanvijayan wrote:
| 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.
| schiffern wrote:
| >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. :)
| kannanvijayan wrote:
| Hah, nice find :)
| schiffern wrote:
| Good show, and I appreciate your sentiment about the
| "messiness" of pi.
|
| There's a unit-converting calculator[0] that supports
| exact rational numbers and will carry undefined variables
| through algebraically. With a little hacking, you can
| redefine degrees in terms in an exact rational multiple
| of pi radians. Pi is effectively being defined as a new
| fundamental unit dimension, like distance.
|
| Trig functions can be overloaded to output an exact
| representation when it detects one of the exact
| trigonometric values[1] eg cos(60deg) = 1/2. It will now
| give output values as "X + Y PI", or you can optionally
| collapse that to an inexact decimal with an eval[]
| function.
|
| That's the closest I got to containing the "messiness" of
| pi. Eventually I hit a wall because Frink doesn't support
| exact square roots, so most exact values would be
| decimals anyway.
|
| Still, I can dream!
|
| [0] https://frinklang.org/
|
| [1]
| https://en.wikipedia.org/wiki/Exact_trigonometric_values
| kannanvijayan wrote:
| I suppose you could have added root two as a fundamental
| as well. I suppose that's another problem with the
| irrationals: two irrationals that aren't linearly related
| by a rational are effectively two fundamentals from each
| others perspective.
|
| It's a sad conclusion - though. Computation exists in the
| countable space. So there is no computationally
| representable symbolic model that can ever algebraically
| capture the reals.
|
| The other thing that came to mind when you mentioned
| root-2 is a similar realization as with pi. That somehow
| a diagonal is not well defined in discrete terms with
| respect to two orthogonal vectors. So here once again,
| you have this weird impedance mismatch between
| orthogonality (a rotational concept) and diagonals (a
| linear concept).
|
| I don't have the formalisms to explore these thoughts
| much further than this.. so it's hard to say whether this
| is just some trivial numerological-like observation or if
| there's something more to it. But it's kinda pleasant to
| think about sometimes.
| jibal wrote:
| > If circles are natural and numbers are natural, then why does
| their relationship seem so unnatural and arbitrary?
|
| It is not in any way unnatural or arbitrary.
|
| However, there are no circles in nature.
|
| > You could imagine some advanced alien civilization, maybe in
| a completely different universe, that isn't even aware of these
| concepts.
|
| I can't actually imagine that ... advancement in the physical
| world requires at least mastery of the most basic facts of
| arithmetic.
|
| > just hoping someone can enlighten me
|
| I suggest that you first need some basic grounding in math and
| philosophy.
| amiga386 wrote:
| 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?
| BobbyTables2 wrote:
| 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.
| amiga386 wrote:
| > 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
| pelorat wrote:
| Wait until you hear about the gluon, the mediator of the
| strong force, which is an excitation in the gluon field, and
| is also the only other particle that is massless and moves at
| C. However unlike the photon the excitation has a really
| short range because gluons interact with gluons and form flux
| tubes between quarks, the further you pull two quarks apart,
| the more energy you need to use, eventually the energy is so
| great that it spawns a new quark from the vacuum.
|
| Compared to EM it's just weird as hell and tbh I don't like
| it.
| notfed wrote:
| > I'm willing to believe elecromagnetic fields are real
|
| No shade intended, but a philosophical conversation is
| unconstructive when it centers around highly ambiguous and
| undefined words. The word "real" does not actually have a
| general meaning until you give it a definition in support of
| your comment. (And surely you will find that if you had a
| definition, you would not need so much "belief" to back up
| your argument.)
| amiga386 wrote:
| I was mostly going down a sciencey path, but "real" is a
| fairly well understood word (part of reality; not
| imaginary).
|
| In terms of philosophy I'm mostly of an empirical bent.
| Things which are observable are real, and things which
| aren't observable directly, but have a observable effect
| that can be repeatedly demonstrated on demand, are real too
| (though they may not be exactly as hypothesised if all we
| can see are their effects). This is how electromagnetism
| and quantum tunnelling can be real at the same time faeries
| aren't.
| eprparadox wrote:
| 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...
| kandel wrote:
| My pet philosophy is that math is real because the objects have
| persistent effects, like with the "if a tree falls in the
| forest..." riddle. Something that isn't real would be a story,
| because things do not have effects in it.
|
| If a function is one-to-one, it has a (right? left? keep
| forgetting which one)-inverse. But if Moshe the imaginary
| forgot the milk, his wife may or may not shout at him,
| whichever way the story teller decides to take the story... So
| a function being one-to-one is real, but Moshe the imaginary
| forgetting the milk isn't.
|
| I like this view when I'm being befuddled by a result,
| especially some ad absurdum argument. I tell myself: this thing
| is true, so if it wasn't we'd just need to look hard enough to
| find somewhere where two effects clash.
| qnleigh wrote:
| I read the Quanta article on this when it came out. They show the
| knots, and they're simple enough that I was almost surprised that
| the counterexample hadn't been found before. But seeing the
| shockingly complicated unknotting procedure here makes it much
| clearer why it wasn't!
|
| It's interesting that you have to first weave the knot around
| itself, which adds many more crossings. Only then do you get a
| the special unknotting that falsifies the conjecture.
| Antinumeric wrote:
| 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.
| deadfoxygrandpa wrote:
| 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
| ealexhudson wrote:
| 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...
| Antinumeric wrote:
| 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.
| iainmerrick wrote:
| 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.)
| gcanyon wrote:
| 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.
| Antinumeric wrote:
| (non-mathematical) Implication doesn't mean certainty,
| which is where I stand with that. But I would posit that
| it (mathematically) implies that joining two knots with
| under to over will _never_ decrease the unknotting number
| from the sum.
| robinhouston wrote:
| I think this is one of those language barrier things. Non-
| mathematicians sometimes say 'obvious' when what they mean is
| 'vaguely plausible'.
| Timwi wrote:
| 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.
| steve_gh wrote:
| Not quite 'obviously', but mathematical folklore has it
| that 'clearly' is used to mark the difficult conceptual
| step in a proof.
| iainmerrick wrote:
| 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.
| jibal wrote:
| Logic fail. The example is not the conjecture. Saying the
| example is obvious is not saying that the conjecture is
| obvious.
| gcanyon wrote:
| 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.
| jibal wrote:
| "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".
| jibal wrote:
| P.S. To clarify:
|
| Saying that the counterexample is _a posteriori_ obvious is
| not saying that the conjecture is _a priori_ obviously
| false.
| pfortuny wrote:
| It happens: once you see the example, it may be trivial to
| understand. The hard thing is to find it.
| James_K wrote:
| 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.
| gus_massa wrote:
| > _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.
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