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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, 2015 at 21:13
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    $\begingroup$ @KenBaker: I think you mean tunnel number =1? $\endgroup$
    – Ian Agol
    Oct 6, 2015 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, 2015 at 3:11
  • $\begingroup$ Thank you very much. But how i find this Knot on this link you posted? $\endgroup$ Oct 6, 2015 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, 2015 at 1:52

2 Answers 2

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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. $\endgroup$ Dec 14, 2015 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, 2015 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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