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New knot theory discovery overturns long-held mathematical assumption

(www.scientificamerican.com)
141 points baruchel | 3 comments | 02 Sep 25 11:35 UTC | HN request time: 0.78s | source
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argomo ◴[04 Sep 25 04:42 UTC] No.45123645[source]▶
Maybe the article is dumbing it down too much, but the conclusion seems unsurprising. Why shouldn't a single unknotting do double-duty in some cases?

It feels akin to the classic trick of joining a tetrahedron to a square pyramid: 4 faces + 5 faces == 5 faces total!

https://m.youtube.com/watch?v=rXIzUtLG2jE

replies(5): >>45124043 #>>45124146 #>>45125679 #>>45125739 #>>45126466 #
Someone ◴[04 Sep 25 10:44 UTC] No.45125739[source]▶
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?
replies(1): >>45126022 #
masterjack ◴[04 Sep 25 11:24 UTC] No.45126022[source]▶
Remarkably there’s really just one way to tie them together, you can always manipulate the knot to move between the different variants
replies(1): >>45127595 #
1. aleph_minus_one ◴[04 Sep 25 14:22 UTC] No.45127595[source]▶
> 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

replies(1): >>45129051 #
2. cottonseed ◴[04 Sep 25 16:29 UTC] No.45129051[source]▶
Those are the same. To see that, just flip over L before performing the connect sum.
replies(1): >>45137800 #
3. cluckindan ◴[05 Sep 25 12:36 UTC] No.45137800[source]▶
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.