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Is there an example of a closed Haken hyperbolic 3-manifold of Heegaard genus 2?

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    $\begingroup$ Try 0-surgery on some hyperbolic knots of tunnel number > 1 and genus > 1. KnotInfo will give a list (indiana.edu/~knotinfo), then check with SnapPy. For example, SnapPy reports that 0 surgery on 6_2 is hyperbolic. $\endgroup$ – Ken Baker Oct 5 '15 at 21:13
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    $\begingroup$ @KenBaker: I think you mean tunnel number =1? $\endgroup$ – Ian Agol Oct 6 '15 at 3:07
  • $\begingroup$ @IanAgol: Yes, of course. That's a typo. And indeed 6_2 has tunnel number = 1. $\endgroup$ – Ken Baker Oct 6 '15 at 3:11
  • $\begingroup$ Thank you very much. But how i find this Knot on this link you posted? $\endgroup$ – Vanderson Lima Oct 6 '15 at 15:34
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    $\begingroup$ @VandersonLima: From indiana.edu/~knotinfo, click on Advanced Search then set Tunnel Number = 1, Three Genus > 1, and Volume > 0 before hitting submit. That should return a list of 135 knots. $\endgroup$ – Ken Baker Oct 7 '15 at 1:52
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Even better, there are hyperbolic surface bundles with Heegaard genus two. These are all described in Jesse Johnson's paper, titled Surface bundles with genus two Heegaard splittings. You will need to use some criterion to recognize pseudo-Anosov maps, however.

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  • $\begingroup$ Thank you for the answer. These are very nice examples indeed. $\endgroup$ – Vanderson Lima Dec 14 '15 at 15:37
  • $\begingroup$ @VandersonLima - After seeing your comment, I thought to check KnotInfo. It says that 57 of the 135 knots (as in Ken Baker's comment) are fibered. Jesse's construction will give you infinitely more examples of the kind you asked for, but they won't all be fillings of knots in the three-sphere. $\endgroup$ – Sam Nead Dec 14 '15 at 20:24
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I found other examples in the literature.

In http://link.springer.com/article/10.1007%2FBF02110720, A. Yu. Vesnin and A. D. Mednykh proved that the Fibonacci manifolds F_{n} have Heegaard genus 2, for n > 2. It is known that for n > 3 these manifolds are hyperbolic.

Moreover the results in http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=2093944&fileId=S0305004100063465, by María Teresa Lozano and Józef H. Przytycki, and http://msp.org/pjm/2000/194-2/p13.xhtml, by Kevin P. Scannell, imply that for n > 5, the manifolds F_n are Haken.

So, this gives infinitely many examples.

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