Comment by gcanyon
6 days ago
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.