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Does there exist a fibering hyperbolic ribbon knot such that the 0 surgery is exceptional? If so does there exist such an example where the result of 0-surgery is Seifert fibered?

I tried looking at knot tables but I'm a bit out of my waters here.

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I had a quick look at Maggie Miller's thesis. It appears that the pretzel knots $(\pm 2, n, -n)$ where $n$ is odd are fibered ribbon knots (and they are also hyperbolic). Moreover, it appears that the $0$-framed surgery is exceptional. I put the ones for $n=3, 5$ into SnapPy, and it couldn’t compute a hyperbolic structure. Small covers have group presentations with commutators, which gives evidence that they are not hyperbolic. Then I created a link $10^3_4$ in SnapPy for which $-1/k, 1/k$ surgery on two components (green and red in the figure) gives the $(-2, 2k+1,-2k-1)$ pretzel knot. SnapPy could not compute the hyperolic structure on $0$-framed surgery on the first component and found a Klein bottle or Mobius strip in it, so this manifold appears to not be hyperbolic. Then large surgeries on it will also not be hyperbolic. The output from SnapPy is not rigorous, but I find it to be reliable, especially on such a small example.

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  • $\begingroup$ Thanks a lot. Do you think the result of zero surgery has a chance of being seifert fibered or at least have vanishing simplicial volume? $\endgroup$
    – ThorbenK
    Commented May 27, 2023 at 13:18
  • $\begingroup$ @ThorbenK I used SnapPy to split along the surface in the 0-surgery on the black component of $10^3_4$, and it gives the link complement $6^3_1$ which is the 3-chain link of volume 5.33... (conjecturally the smallest 3-cusped manifold). So I think this manifold is obtained from $6^3_1$ by gluing in a Klein bottle and has non-zero simplicial volume, so most surgeries will too. On the other hand, with $n=3$, it seems like the manifold might have simplicial volume 0. SnapPy doesn't find any splitting surfaces, which might mean it is a small Seifert-fibered space? $\endgroup$
    – Ian Agol
    Commented May 27, 2023 at 16:31
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    $\begingroup$ Fiddling around a bit more, it appears that $0$-surgery on the $(-2,-3,3)$ pretzel knot is not Seifert-fibered. If it were, it should be based on $H^2\times R$ or $SL(2,R)$ geometry. In either case, there is a cover with torsion in $H_1$ of rank $\leq 1$. But SnapPy computes covers with torsion of rank $>1$, so it seems like it's not Seifert-fibered. $\endgroup$
    – Ian Agol
    Commented May 31, 2023 at 16:23
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    $\begingroup$ A quick comment: regina can usually recognize non-hyperbolic manifolds. So by sending the SnapPy triangulation of the filled manifold to regina it should be possible to recognize the manifold explicitly. If I am not mistaken the pretzel knot P(2,3,-3) is the census knot m222 and P(2,5,-5) is o9_35240. SnapPy can confirm that. For the census knots Dunfield has already classified all exceptional fillings (arxiv.org/abs/1812.11940). By checking his list one sees for example that the 0-filling of the second pretzel knot is a SFS over the Möbius strip glued to m043. $\endgroup$
    – Marc Kegel
    Commented Jun 5, 2023 at 10:48
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    $\begingroup$ By recomputing that slopes we see that the 0-filling of P(-2,-3,-3) is a graph manifold. In regina's notation: SFS [D: (2,1) (2,1)] U/m SFS [D: (3,1) (3,2)], m = [ 0,1 | 1,0 ] $\endgroup$
    – Marc Kegel
    Commented Jun 5, 2023 at 10:55

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