12
$\begingroup$

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.

Improve this question
$\endgroup$

3 Answers 3

Reset to default
13
$\begingroup$

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.

Improve this answer
$\endgroup$
11
$\begingroup$

The existence of such examples is proved in a preprint of Jesse Johnson. He proves that for any $n$, there is a tunnel number 1 knot which has bridge number at least $n$. A tunnel number one knot complement has a Heegaard surface of genus 2, which splits the knot complement up into a genus 2 handlebody and a genus 2 compression body with a torus boundary corresponding to the boundary of a regular neighborhood of the knot. Thus tunnel number 1 knots are 2-generator (and in fact with 1 relation), so this answers your question.

Improve this answer
$\endgroup$
4
  • 4
    $\begingroup$ Steven Sivek's answer is much more elementary than the reference I gave, in fact torus knots have bridge numbers $\to \infty$, yet are tunnel number 1. The point of Jesse's construction is that he gave hyperbolic examples with this property. $\endgroup$
    – Ian Agol
    Commented Jul 24, 2012 at 22:10
  • $\begingroup$ Okay, I'll give Steven the checkmark then. $\endgroup$
    – Ryan Budney
    Commented Jul 24, 2012 at 22:11
  • $\begingroup$ Where do the lower bounds on the bridge numbers of torus knots come from? $\endgroup$
    – Ryan Budney
    Commented Jul 24, 2012 at 22:39
  • 4
    $\begingroup$ A little Googling later I found in Jennifer Schulten's paper "Brige numbers of torus knots" that the bridge number of the $(p,q)$ torus knot is the minimum of $p$ and $q$, nice. And this is due to Schubert. $\endgroup$
    – Ryan Budney
    Commented Jul 24, 2012 at 22:43
9
$\begingroup$

It is not a problem, but this is a problem.

A question posed by Cappell and Shaneson asks whether the bridge number of a given link equals the minimal number of meridian generators of its group.

http://ccms.claremont.edu/topology-seminar/Bridge-numbers-links-and-meridian-ranks-link-groups

Improve this answer
$\endgroup$
3
  • $\begingroup$ This is also problem 1.11 in Kirby's list, attributed therein to Cappell and Shaneson. $\endgroup$
    – dvitek
    Commented Oct 9, 2013 at 4:30
  • $\begingroup$ Chris Cornwell obtained lower bounds on the meridional rank of certain pretzel knots, showing they are equal to the bridge number: arxiv.org/abs/1303.4943 $\endgroup$
    – Ian Agol
    Commented Mar 31, 2015 at 3:32
  • $\begingroup$ The statement that "the bridge number of a given link equals the minimal number of meridian generators of its group" is known as the meridional rank conjecture. See e.g. dx.doi.org/10.4310/CAG.2020.v28.n2.a2 $\endgroup$
    – Lukas Lewark
    Commented Dec 22, 2021 at 7:54

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .