# A knot problem

1. Jan 16, 2005

### spill

This is a problem that's been troubling me recently. I'm neither a boy scout nor a knot theoretician, so I'm not sure how to progress.

Is it possible for three pieces of string to be tied together without the knot being topologically equivalent to a knot tying two of the pieces together with the third string tied either around one of the other pieces or around the first knot itself?

Oh dear, that wasn't very clear. Let's try again.

Is it possible to construct a knot from three pieces of string where removing any of the pieces will cause the knot to collapse into an unknot?

Help appreciated.

2. Jan 16, 2005

### hypermorphism

Since a knot cannot self-intersect, removing an arc from a knot will make it a curve that is not a knot. If you mean remove an arc and close the curve, then there are many obvious choices of making an unkot from a nontrivial knot. If you meant 3 links and the resulting 2 links collapsing into the unlink, then the Borromean rings is an example of one such link. Such links are called http://www.cs.ubc.ca/nest/imager/contributions/scharein/brunnian/brunnian.html and can have any number of components.
I've probably misunderstood your question, though. :)

Last edited by a moderator: Apr 21, 2017
3. Jan 18, 2005

### spill

Brunnian links - exactly what I was looking for. Thanks!