Comment by Antinumeric

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.