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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?

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  • $\begingroup$ 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
    Commented Apr 29, 2016 at 4:16

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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.

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  • $\begingroup$ Fantastic, Ian. Thank you very much. $\endgroup$
    – charlie katerba
    Commented Apr 29, 2016 at 14:42
  • $\begingroup$ You're welcome @Charlie (Frohman? Livingston? Trotter?). $\endgroup$
    – Ian Agol
    Commented Apr 30, 2016 at 2:40
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    $\begingroup$ 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 katerba
    Commented Apr 30, 2016 at 3:28
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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 In unpublished work, Ken Baker showed that certain Berge knot complements could be large: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.

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