Comment by Antinumeric
5 days ago
I'm not saying I could have come up with the example. I'm saying looking at the example, and seeing how the two unders are connected togther, and the two overs connected together, makes it obvious that there is more freedom to move the knot around. And that freedom, at least to me, is intuitively connected to the unknotting number.
And that is why the mirror image had to be taken - you need to make sure that when you join it is over to over and under to under.
You’re getting a lot of pushback here, but I have to say, your intuition makes sense to me too.
When you’re connecting those two knots, it seems like you have the option of flipping one before you join them. It does seem very plausible that that extra choice would give you the freedom to potentially reduce the knotting number by 1 in the combined knot.
(Intuitively plausible even if the math is very, very complex and intractable, of course.)
But this implies that a simple 1-knot might completely undo itself if you join it to its mirror. Which I assume people have tried, and doesn't work. Likewise with 2's, 3's etc.
It seems intuitively obvious that there is something deeper going on here that makes these two knots work, where (presumably) many others have failed. Or more interestingly to me, maybe there's something special about the technique they use, and it might be possible to use this technique on any/many pairs of knots to reduce the sum of their unknotting numbers.
(non-mathematical) Implication doesn't mean certainty, which is where I stand with that. But I would posit that it (mathematically) implies that joining two knots with under to over will never decrease the unknotting number from the sum.