Linking Venn Diagrams to Paper Clip Chains

Hello! Thanks for reading and keeping up with the posts here.


So far, in the course of these posts, we've only been talking about or single knotted loops of string.

Today, we'll be talking about links, or multiple knots mixed together.

If you have ever made paper clips chains or have drawn a Venn Diagram, think of it as something like that, except with the loops of knots (Figure 1, 2).

Figure 1 - A triple diagram. (Or, as a link, famously called the Borromean Rings.)

Figure 2 - A paper clip chain - Links are like this in that they are looped in eachother.

The diagram in Figure is be an example of the Borromean Rings link. Another example of a link, would be the Whitehead Link below

Figure 3 - The Whitehead Link is made up of two separate loops.

We say a link has n components if it is made up of n different parts. For example, the Borromean Rings link in Figure 1 would be composed of three components.



Links can be classified as splittable and non-splittable. Splittable links are those you can twist so that the pieces become separate.

Think of this as taking two rubber bands and trying to roll them together (we've all done it...). No matter how you try to push them together, you can still pull them apart (move them around) so that they're back to two separate rubber bands.

Splittable links are kind of like this, except, instead of just rubber bands, you can take knots and try to smush them together. The figure below shows the two examples.

Figure 4  - Two different splittable links. 

We mostly focus on unsplittable links (such as the Whitehead Link in Figure 4), as splittable links can be pulled apart and considered as two separate knots.

That's it for now (as this has been a pretty long post), and we'll continue after a short intermission!