Maybe I'm dumb, but they have two knots that have a number of 3, one is the mirror of the other. They were hoping that it would add up to six, but it only adds to 5.

Wouldn't this mean that there is a sort of "negative" number implied here? That one knot is +2/+1 and that the other knot is +2/-1, and that their measure (the unknotting number) is only the sum of the abs()?

An analogy might be how if you mix together water and alcohol, you get a solution with less volume than the sum of the volumes. That doesn't mean that there's "negative" volume, just that the volume turns out to be sub-additive due to an interaction of specific characteristics of the liquids. Somehow, some connect sums of particular knots enable possibilities that let it more easily be unknotted.

I spent the better part of the summer during grad school trying to prove additivity of unknotting numbers. (I'll mention that it's sort of a relief to know that the reason I failed to prove it wasn't because I wasn't trying hard enough, but that it was impossible!)

One approach I looked into was to come up with some different analogues of unknotting number, ones that were conceptually related but which might or might not be additive, to at least serve as some partial progress. The general idea is represent an unknotting using a certain kind of surface, which can be more restrictive than a general unknotting, and then maybe that version of unknotting can be proved to be additive. Maybe there's some classification of individual unknotting moves where when you have multiple of them in the same knotting surface, they can cancel out in certain ways (e.g. in the classification of surfaces, you can always transform two projective planes into a torus connect summand, in the presence of a third projective plane).

Connect summing mirror images of knots does have some interesting structure that other connect sums don't have — these are known as ribbon knots. It's possible that this structure is a good way to derive that the unknotting number is 5. I'm not sure that would explain any of the other examples they produced however — this is more speculation on how might someone have discovered this counterexample without a large-scale computer search.

I don't think it implies that one of the knots has a negative number. It proves that when you add knots together you can't just add together their unknotting numbers and expect to get a correct answer. The article mentions "unpredictability of the crossing change" as a source of this issue (if I am reading that statement correctly).

Basically the unknotting number combes from how the string crosses itself and when you add two (or more?) knots together you can't guarantee that the crossings will remain the same, which makes a kind of intuitive sense but is extremely frustrating when there isn't a solid mathematical formula that can account for that.

That would certainly be interesting, though I don't know of any matching definition in knot theory. There is a notion of "positive" and "negative" crossings, so you could define the positive and negative unknotting numbers by asking how many of each you have to swap. Unfortunately, in their example, all torus knots can be drawn with all positive or all negative crossings.

Not exactly. Orientation is a bitch, just make a mobius strip anc cut it across its middle, and do it again… magic.

I think you are reaching more for imaginary numbers?

I could see trying to fit this with surreal numbers, as well. Would be fitting, as I think Conway was big into knots?

Regardless, no, not dumb. Numerically modelling things is hard, it turns out. :D