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?

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masterjack 5 days ago | parent [-]

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

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aleph_minus_one 5 days ago | parent [-]

> 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

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cottonseed 5 days ago | parent [-]

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

	▲	cluckindan 4 days ago \| parent [-]
		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.