We say a knot $K$ in $S^3$ is small if its exterior contains no closed properly embedded incompressible surfaces and we say $K$ is large otherwise.

Does anyone know of an example of a large hyperbolic knot $K$ such that $\pi_1( S^3 - K)$ has rank 2? All of the examples of large knots I can think of (4-strand pretzel knots and some knots whose complements contain embedded quasi-Fuchsian surfaces for instance) seem to have fundamental groups with rank 3 or larger.

How about, more generally, examples of large hyperbolic 3-manifolds with rank two fundamental group?

  • $\begingroup$ For the answer to your second question, one may find hyperbolic 3-manifolds with Heegaard genus 2 and $b_1 >0$, which are thus large with rank 2 fundamental group. I suspect knot examples exist, but I'll have to think about it. $\endgroup$
    – Ian Agol
    Apr 29, 2016 at 4:16

2 Answers 2


Mario Eudave-Muñoz constructed examples of tunnel number one hyperbolic knots containing closed incompressible surfaces. See Theorem 8.1:

Mario Eudave-Muñoz, MR 1719999 Incompressible surfaces in tunnel number one knot complements, Topology Appl. 98 (1999), no. 1-3, 167--189.

  • $\begingroup$ Fantastic, Ian. Thank you very much. $\endgroup$
    – Charlie
    Apr 29, 2016 at 14:42
  • $\begingroup$ You're welcome @Charlie (Frohman? Livingston? Trotter?). $\endgroup$
    – Ian Agol
    Apr 30, 2016 at 2:40
  • 1
    $\begingroup$ None of the above. My last name is Katerba - I'm a grad student at the University of Montana studying under Eric Chesebro. $\endgroup$
    – Charlie
    Apr 30, 2016 at 3:28

Here is another set of (possibly overlapping) examples which also might be interesting. In unpublished work, Ken Baker showed that certain Berge knot complements could be large:

Kenneth L. Baker, Closed essential surfaces in the complements of large volume Berge knots. arXiv preprint math/0509082 (2005).

Berge knot complements necessary have two generator fundamental group, because they are tunnel number one as well.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.