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.

counter-example results are always fun

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.

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.

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 =

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?

They only had research mathematicians working on the problem. Until now they didn't have HN commenters. So work went very slowly.

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.

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.

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.