Knot theory: pretzel knot

Question

Prove that pretzel knot $K(p_1,p_2,p_3,\dots,p_n)$ with all $p_i >0$ is an alternating knot or link?

I think since all $p_i$'s are positive, the sign has a lot to do with it but how to prove it is my issue

Answer 1

You can probably convince yourself that this is true by drawing a few examples (try $K(2,3,5)$, for instance); let's start somewhere on the link diagram and follow it around to see what can happen. Clearly, as long as you are staying within one of the tangles that make up the link, you'll alternate between over- and under-crossings, so the only interesting thing happens when you leave one of the tangles. Now, depending on the parity of the $p_i$ that you are following, you'll go either to the tangle to the left or the one to the right; correspondingly, you'll come from either an undercrossing or an overcrossing respectively, and the assumption that all $p_i$ are positive will ensure that if you go to the left, you'll meet an overcrossing, and if you go to the right, you'll meet an undercrossing.