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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?

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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":

enter image description here

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.

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