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.

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.