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 =
magicalhippo · 9h ago
> '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.
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.
jxbdbdbd · 7h ago
That is not necessarily true. Knot theory is quite niche, maybe nobody before tried bruteforcing counter examples
trueismywork · 2h ago
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.
nyeah · 4h ago
It's not necessarily true. But it's pretty likely. It's worth considering as a possibility.
nyeah · 5h ago
They only had research mathematicians working on the problem. Until now they didn't have HN commenters. So work went very slowly.
tpoacher · 2h ago
bwahahah, loved this comment.
Hnrobert42 · 4h ago
Do you feel this substantively contributes to the conversation?
nyeah · 4h ago
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.
Hnrobert42 · 4h ago
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.
cyphar · 3h ago
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.
nyeah · 3h ago
If anybody is reading this, please hit "parent" a few times to see what everybody actually said.
tpoacher · 2h ago
I do. It gave me a good ol' chuckle. That's a great contribution to the conversation right there!
Someone · 6h ago
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?
masterjack · 6h ago
Remarkably there’s really just one way to tie them together, you can always manipulate the knot to move between the different variants
aleph_minus_one · 3h ago
> 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
cottonseed · 57m ago
Those are the same. To see that, just flip over L before performing the connect sum.
viraptor · 11h ago
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?)
lilyball · 10h ago
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.
viraptor · 10h ago
> crossing changes (which don't change the number of crossings
Ok, that explains the search space explosion. Thanks for explaining!
cottonseed · 54m ago
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.
adgjlsfhk1 · 10h ago
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.
d--b · 8h ago
This is when you read articles like these that you realize how great the articles on quanta magazine are.
“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.
cka · 17h ago
Wow, this problem has been around for a long time. Exciting to see this finally figured out.
NooneAtAll3 · 14h ago
counter-example results are always fun
binary132 · 13h ago
I’m curious what specific conclusions this may undo.
fjfaase · 12h ago
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.
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
> 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 =
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
Maybe you meant to ask something else. But you asked about substance.
> 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.
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
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.
Ok, that explains the search space explosion. Thanks for explaining!
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.
“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.