Question about closing a chain of elastic bands -- and hatching eggs

[Alex Galicki]

You're correct, it's possible to create a "loop" without a Hopf link.
And if the goal is to have only such "connections" as on your picture,
there should be no Hopf links at all.

It seems to me that if there are no restrictions as to how to "cut and
repair" links, the problem becomes much more general than I thought:
how to distinguish a trivial link from a non-trivial one. In that
case, Jones polynomials certainly won't suffice. They might be
sufficient for distinguishing Brunnian link, I don't know.

However, after a wonderful session of procrastination, I found the
following paper: https://arxiv.org/pdf/1809.10334.pdf

Where it says: ``The sublink problem asks ”Given diagrams of two
links, is there a sublink of the first that is isotopic to the
second?” Lackenby showed that this problem is NP–hard using a Karp
reduction (see Section 3.1.2 in [7]) from the HAMILTONIAN PATH problem
[14]. Here we examine the UNLINK AS A SUBLINK problem, in which the
second link is an unlink. UNLINK AS A SUBLINK: given a diagram for a
link L and a positive integer k, is there a k-component sublink of L
that is an unlink?

Theorem 1. UNLINK AS A SUBLINK is NP–complete.''

So it seems the problem is definitely computable, possibly NP-complete
(if the arxiv paper is to be believed).

On Fri, 10 Sept 2021 at 12:26, Timothy Y. Chow <tchow at math.princeton.edu> wrote:
>
> Alex Galicki wrote:
> > However, if we say that "ring" means the end result of the procedure you
> > have described ("if the last band added is cut before the chain is
> > closed and the cut repaired afterwards"), then such a ring always
> > contains a Hopf link.
>
> Perhaps Aaron should clarify, but this doesn't sound right to me.  When
> you cut and repair, what you want to end up with is something that looks
> like the picture shown here:
>
> https://i.ytimg.com/vi/gntXhYfunC4/maxresdefault.jpg
>
> That's not a Hopf link.  Another way to see it is, in Aaron's
> configuration, if you delete any single rubber band, then the whole thing
> comes apart (a "Brunnian link").  So if there are at least three rubber
> bands, then there cannot be a Hopf link---choose a rubber band not
> involved in the Hopf link and delete it, and the Hopf link remains linked.
>
> Googling around, I found this page, which seems to indicate that the Jones
> polynomial suffices to distinguish it from an unlink:
>
> http://katlas.math.toronto.edu/wiki/%22Rubberband%22_Brunnian_Links
>
> Tim