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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?


Remarkably there’s really just one way to tie them together, you can always manipulate the knot to move between the different variants


> Remarkably there’s really just one way to tie them together

I would rather assume (but knot theorists shall correct me if I'm wrong) that there exist two ways of tying them together:

Cut knots K, L at some point; denote the loose ends by K1, K2, L1, L2.

- Option 1: connect K1 <-> L1, K2 <-> L2

- Option 2: connect K1 <-> L2, K2 <-> L1


Those are the same. To see that, just flip over L before performing the connect sum.


If they are the same, the mirrored double-chiral knot from the article would have identical properties even if one of the knots wasn’t mirrored.




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