According to the actual paper (https://arxiv.org/pdf/2506.24088), it has been an open conjecture since at least 1977. The quote:
> Unknotting number has long been conjectured to be additive under
connected sum; this conjecture is implicit in the work of Wendt, in one
of the first systematic studies of unknotting number [37]. It is unclear
when and where this was first explicitly stated; most references to it
call it an ‘old conjecture’. It can be found in the problem list of Gordon
[13] from 1977 and in Kirby’s list [16].
'Additive' here means that if u(K1) is defined as the unknotting number of the knot K1, and u(K1#K2) the unknotting number of the knots K1 and K2 joined together, then u(K1#K2) = u(K1) + u(K2). It is this that has (assuming the paper is correct) been proven false. A deceptively simple property!
edit: I initially incorrectly had a ≤ sign instead of =
> 'Additive' here means that if u(K1) is defined as the unknotting number of the knot K1, and u(K1#K2) the unknotting number of the knots K1 and K2 joined together, then u(K1#K2) ≤ u(K1) + u(K2).
Kinda like the triangle inequality[1] of knots?
I recall the triangle inequality was useful for several cases in Uni, if so I guess I can see it might be a similarity useful inequality in knot theory.
We have huge data about knots in protein folding. Given that the proof is a counterexqmple, if it was easy, it should have been observed already in data I feel.
Yes. I feel that way very strongly. What contains no substance is a discussion of how we are smarter about knot theory than the knot theorists ... without even connecting to what makes the problem difficult.
Maybe you meant to ask something else. But you asked about substance.
GP explicitly stated they might be misunderstanding. If you see how they misunderstood, perhaps you could explain. An appeal to authority isn't much of an explanation.
> Maybe the article is dumbing it down too much, but the conclusion seems unsurprising. Why shouldn't a single unknotting do double-duty in some cases?
is them "explicitly stat[ing] they might be misunderstanding"? At best they said that the article is at fault for oversimplifying the topic.
Author of the comment you're quoting, and it is indeed my roundabout way of suggesting I'm missing something.
Clearly, I'm not a knot theory expert, but the way the article presents it makes me wonder what extra nuance motivated the original (now falsified) conjecture.
I also do not understand the intuition behind the assumption. To tie two knots together, you have to make a cut in both of them, and you have two ways to tie them together again. Doesn’t that introduce some opportunity to get rid of some complexity of the knots?
“Unknotting number is not additive under connected sum” (2025 v1)
> We give the first examples of a pair of knots K1,K2 in the 3-sphere for which their unknotting numbers satisfy u(K1#K2)<u(K1)+u(K2) . This answers question 1.69(B) from Kirby's problem list, "Problems in low-dimensional topology", in the negative.
Is this something people have been actively trying to disprove? The example provided seems to not be hard to bruteforce - given it's only 5 moves. Does anyone know why there's no older counter example? (Or am I totally underestimating how the number of options explodes in 5 moves?)
It's not just 5 moves. It's 5 crossing changes (which don't change the number of crossings, they just change the order of the strings in a crossing). Unknotting also involves moving the strings around to add or remove crossings, without performing crossing changes (if you take a loop and twist it into a figure eight, you've moved the strings and created a crossing but you haven't cut the strings and performed a crossing change).
If you look at the preprint paper, the knot it starts with has 14 crossings, but they actually move the strings around to end up with 20 crossings prior to performing the first 2 crossing changes in the unknotting sequence. So the potential space for moves here is actually rather large.
1: knot theory is somewhat obscure. it generally only comes up in undergrad in a topology class for a week or two so there aren't a ton of people interested
2. It's 5 cuts on a joining of 2 knots with 6 crossings. it's brute forcable, but not trivially (i.e. you have to code it up and possibly wait a while)
3. for conjectures that feel intuitively true more effort goes into finding the proof than looking for a counterexample that feels unlikely to exist.
You cannot bruteforce this. Exhibiting a unknotting of K with n moves only gives you an upper bound u(K) <= n. Proving u(K) = n is an entirely different matter.
It is about the problem of untying knots. For many complex knots it is not know what is the minimal number of steps that are needed to unty it. There was this idea that if a complex knot consisted of two knots for which it is known, that the number would be equal to the sum of the number of steps of the two knots. The article shows that that is not true by showing an example of a knot where the number is one less. This shows that there is no easy route for finding the number for ever larger knots.
Maybe the article is dumbing it down too much, but the conclusion seems unsurprising. Why shouldn't a single unknotting do double-duty in some cases?
It feels akin to the classic trick of joining a tetrahedron to a square pyramid: 4 faces + 5 faces == 5 faces total!
https://m.youtube.com/watch?v=rXIzUtLG2jE
According to the actual paper (https://arxiv.org/pdf/2506.24088), it has been an open conjecture since at least 1977. The quote:
> Unknotting number has long been conjectured to be additive under connected sum; this conjecture is implicit in the work of Wendt, in one of the first systematic studies of unknotting number [37]. It is unclear when and where this was first explicitly stated; most references to it call it an ‘old conjecture’. It can be found in the problem list of Gordon [13] from 1977 and in Kirby’s list [16].
'Additive' here means that if u(K1) is defined as the unknotting number of the knot K1, and u(K1#K2) the unknotting number of the knots K1 and K2 joined together, then u(K1#K2) = u(K1) + u(K2). It is this that has (assuming the paper is correct) been proven false. A deceptively simple property!
edit: I initially incorrectly had a ≤ sign instead of =
> 'Additive' here means that if u(K1) is defined as the unknotting number of the knot K1, and u(K1#K2) the unknotting number of the knots K1 and K2 joined together, then u(K1#K2) ≤ u(K1) + u(K2).
Kinda like the triangle inequality[1] of knots?
I recall the triangle inequality was useful for several cases in Uni, if so I guess I can see it might be a similarity useful inequality in knot theory.
[1]: https://en.wikipedia.org/wiki/Triangle_inequality
I incorrectly had a ≤ instead of =, my apologies.
Ah, no worries. So strictly additive.
If a lot of very smart people didn’t find a single example in all the years knot theory has existed, it obviously is not that obvious.
That is not necessarily true. Knot theory is quite niche, maybe nobody before tried bruteforcing counter examples
We have huge data about knots in protein folding. Given that the proof is a counterexqmple, if it was easy, it should have been observed already in data I feel.
It's not necessarily true. But it's pretty likely. It's worth considering as a possibility.
They only had research mathematicians working on the problem. Until now they didn't have HN commenters. So work went very slowly.
bwahahah, loved this comment.
Do you feel this substantively contributes to the conversation?
Yes. I feel that way very strongly. What contains no substance is a discussion of how we are smarter about knot theory than the knot theorists ... without even connecting to what makes the problem difficult.
Maybe you meant to ask something else. But you asked about substance.
GP explicitly stated they might be misunderstanding. If you see how they misunderstood, perhaps you could explain. An appeal to authority isn't much of an explanation.
Which part of this comment:
> Maybe the article is dumbing it down too much, but the conclusion seems unsurprising. Why shouldn't a single unknotting do double-duty in some cases?
is them "explicitly stat[ing] they might be misunderstanding"? At best they said that the article is at fault for oversimplifying the topic.
Author of the comment you're quoting, and it is indeed my roundabout way of suggesting I'm missing something.
Clearly, I'm not a knot theory expert, but the way the article presents it makes me wonder what extra nuance motivated the original (now falsified) conjecture.
If anybody is reading this, please hit "parent" a few times to see what everybody actually said.
I do. It gave me a good ol' chuckle. That's a great contribution to the conversation right there!
I also do not understand the intuition behind the assumption. To tie two knots together, you have to make a cut in both of them, and you have two ways to tie them together again. Doesn’t that introduce some opportunity to get rid of some complexity of the knots?
Remarkably there’s really just one way to tie them together, you can always manipulate the knot to move between the different variants
> Remarkably there’s really just one way to tie them together
I would rather assume (but knot theorists shall correct me if I'm wrong) that there exist two ways of tying them together:
Cut knots K, L at some point; denote the loose ends by K1, K2, L1, L2.
- Option 1: connect K1 <-> L1, K2 <-> L2
- Option 2: connect K1 <-> L2, K2 <-> L1
Those are the same. To see that, just flip over L before performing the connect sum.
If they are the same, the mirrored double-chiral knot from the article would have identical properties even if one of the knots wasn’t mirrored.
[dead]
This is when you read articles like these that you realize how great the articles on quanta magazine are.
Gem of an old video going over the basics of knot theory. https://www.youtube.com/watch?v=QcLfb0PhfO0
It was made on a supercomputer from the 90's
Without paywall: https://www.removepaywall.com/search?url=https://www.scienti...
Linked paper: https://arxiv.org/abs/2506.24088
“Unknotting number is not additive under connected sum” (2025 v1)
> We give the first examples of a pair of knots K1,K2 in the 3-sphere for which their unknotting numbers satisfy u(K1#K2)<u(K1)+u(K2) . This answers question 1.69(B) from Kirby's problem list, "Problems in low-dimensional topology", in the negative.
Is this something people have been actively trying to disprove? The example provided seems to not be hard to bruteforce - given it's only 5 moves. Does anyone know why there's no older counter example? (Or am I totally underestimating how the number of options explodes in 5 moves?)
It's not just 5 moves. It's 5 crossing changes (which don't change the number of crossings, they just change the order of the strings in a crossing). Unknotting also involves moving the strings around to add or remove crossings, without performing crossing changes (if you take a loop and twist it into a figure eight, you've moved the strings and created a crossing but you haven't cut the strings and performed a crossing change).
If you look at the preprint paper, the knot it starts with has 14 crossings, but they actually move the strings around to end up with 20 crossings prior to performing the first 2 crossing changes in the unknotting sequence. So the potential space for moves here is actually rather large.
> crossing changes (which don't change the number of crossings
Ok, that explains the search space explosion. Thanks for explaining!
I think this is a combination of things.
1: knot theory is somewhat obscure. it generally only comes up in undergrad in a topology class for a week or two so there aren't a ton of people interested
2. It's 5 cuts on a joining of 2 knots with 6 crossings. it's brute forcable, but not trivially (i.e. you have to code it up and possibly wait a while)
3. for conjectures that feel intuitively true more effort goes into finding the proof than looking for a counterexample that feels unlikely to exist.
You cannot bruteforce this. Exhibiting a unknotting of K with n moves only gives you an upper bound u(K) <= n. Proving u(K) = n is an entirely different matter.
Wow, this problem has been around for a long time. Exciting to see this finally figured out.
counter-example results are always fun
I’m curious what specific conclusions this may undo.
It is about the problem of untying knots. For many complex knots it is not know what is the minimal number of steps that are needed to unty it. There was this idea that if a complex knot consisted of two knots for which it is known, that the number would be equal to the sum of the number of steps of the two knots. The article shows that that is not true by showing an example of a knot where the number is one less. This shows that there is no easy route for finding the number for ever larger knots.
Yes I got that, I meant whether there are other theorems or conclusions that would be disproven or altered by disproving this hypothesis.