Bridge index of Pretzel link

Question

I was studying the bridge number of various kinds of links.I have heard there is some correlation between the bridge index of a Pretzel link and it's representation.Can anyone please explain it or suggest me some material where I can find the correlation between these two?

Answer 1

This question becomes much more clear if you are using a certain definition of bridge index. For example, in Rolfsen, he defines bridge index to be least number of overarcs of any diagram of a knot. This is a fine definition for some purposes, but very hard to see how it is connected to pretzel links.

The definition you should use in this case is the following:

The bridge number of a knot is the minimum number of local maximums 
(or minimums) over all diagrams. 

So, if you take this definition, you immediately see that a $n$-pretzel link has a bridge index of at most $n$. This can be seen since one usually has a picture of a pretzel link that looks something like this.

Where in the large box, we have only descending arcs from top to bottom. So the diagram has at most a bridge index of $n$, one for each semi circle at the top (or bottom). Off hand, I am not sure about when the bridge index equals the pretzel index. Perhaps someone else knows that result, but I would not be surprised if in general there is a class of examples where the bridge index is strictly less than the pretzel index.