Given a knot in the 3-sphere in Bridge Position you can find a presentation for the fundamental group of the complement (a Wirthinger presentation) containing $n$ generators and $n-1$ relators, where $n$ is the number of bridges in the presentation.
The bridge index of the knot is the minimal number of bridges it takes to present the knot. Is that the same as the minimal number of generators among all presentations of the fundamental group of the knot complement?
I imagine this is well known but after a while of Googling around and looking through Neuwirth's "Knot Groups" book I haven't found where this question is addressed.
Best Answer
The (p,q) torus knot has a presentation with two generators, namely $\langle x,y \mid x^p = y^q\rangle$, but if $p,q>2$ then it's non-alternating and so it must have bridge index greater than 2.