# If L is a split link then L has split link diagram. Is this true?

I am reading the book "An introduction to knot theory" by Lickorish. Following are the definitions at the beginning of the chapter "Geometry of Alternating Links".

A link $$L$$ in $$\mathbb{S}^3$$ having at least two components is a split link if there is a $$2$$-sphere in $$\mathbb{S}^3 - L$$, separating $$\mathbb{S}^3$$ into two balls each of which contains a component of $$L$$.

A link diagram $$D$$ in $$\mathbb{S}^2$$ is a split diagram if there is a simple closed curve in $$\mathbb{S}^2 - D$$ separating $$\mathbb{S}^2$$ into two discs each containing part of $$D$$.

The author has written that the main aim of the chapter is to prove the following theorem:

• Suppose a link $$L$$ has an alternating diagram $$D$$. Then $$L$$ is a split link if and only if $$D$$ is a split diagram.

I am thinking why the above result is not true in general. Are there examples of split links such that none of their knot diagrams are split diagrams. I am not able to construct such examples, can someone help?

## 1 Answer

Theorem: Suppose a link L has an alternating diagram D. Then L is a split link if and only if D is a split diagram.

Note that the $$(\Leftarrow)$$ direction of this theorem is trivial. If you have a split diagram... well, then you're a split link for sure. So the content of the theorem is that if someone hands you an alternating diagram of a split link, then it will look obviously like a split link to you. So the more general statement is

(false) Theorem: Suppose a link L has a diagram D. Then L is a split link if and only if D is a split diagram.

This is not true in general, though. I can hand you a non-alternating diagram of a split link and it might take you a while to figure out that it's actually split. Here's one that might not take you so long, but is a counter-example to the "more general theorem":

Are there examples of split links such that none of their knot diagrams are split diagrams?

No. If you have a split link, then you can separate each component from the others with a 2 sphere bubble. Set your bubbles nicely in a row and then project in a convenient direction. This is a split diagram.