When a 3-strand pretzel link is a knot?

Question

I'm working on an article about the 3-strand pretzel links and I came across this remark :

$P(p_1,p_2,p_3)$ is a knot if and only if none of two $p_i'$s are even.

So I thaught that the the number of even $p_i'$s is the number of components.

My question is how to prove the remark.

Thanks in advance.

Answer 1

This can be done in a case by case analysis on the diagrams. If you only care about pretzel knots with 3 twisted regions, then you only have to have a few cases, based on whether there are an even or odd number of twists.

Also, you can further reduce your work by appealing to the other question you asked, so you don't have to worry about where the even ones are in the diagram.

Good luck.