Is this knotted graph knotted?

Asked Oct 07 '20 at 17:34

Active Oct 07 '20 at 17:34

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Above, I've drawn a knotted 3-valent graph. I suspect that it's not isotopic to the "unknotted" version below, but I'm not sure. Is it?

I know about fundamental groups of knot complements, but it seems like that's probably more work than necessary. The easiest way would be to show that it's the companion of some simple knot, but I haven't been able to get that to work.

Is this knotted (by which I mean, is that not-equals sign in the drawing correct), and if so, is there a good way to prove that?

asked Oct 07 '20 at 17:34

Akiva Weinberger

25,412

It seems to be related to the so-called "Hopf handcuffs". – Akiva Weinberger Oct 07 '20 at 17:37
Specifically, you can obtain it by doing this transformation to the Hopf handcuffs. Maybe there's a lemma that says that that transformation preserves knottedness? – Akiva Weinberger Oct 07 '20 at 17:40
(We can iterate that transformation as many times as we like - probably all those iterations are knotted. A fundamental group computation seems kinda inevitable now…) – Akiva Weinberger Oct 07 '20 at 17:52
I vaguely remember Rolfsen's book having a section on an iterated "chain" transformation (used to describe Antoine's necklace). Unfortunately I don't have that book with me so I can't reference it to see how he did it, but maybe we need a similar strategy. – Akiva Weinberger Oct 07 '20 at 17:59
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Ah. At the end of this paper, the Jones polynomial is used to show that this is, in fact, knotted. – Akiva Weinberger Oct 07 '20 at 19:14
Here's a much better reference. They consider "knotted handlebodies", which are these knotted graphs (or "spatial graphs") up to IH moves (moves that replace an H shape with a sideways H shape). They make a table of the smallest knotted handlebodies. The first graph in my image is 6_10; the second is 0_1 (up to an IH move). – Akiva Weinberger Oct 07 '20 at 23:35
(Paywall or university account only, apparently) – Akiva Weinberger Oct 07 '20 at 23:47
(or, y'know, sci-hub) – Akiva Weinberger Jun 17 '22 at 03:56

Is this knotted graph knotted?

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