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For $K$ a hyperbolic knot in $S^3$, a rational number $p/q$ is an exceptional slope if the manifold $M_{p/q}$ obtained from $(p,q)$-Dehn surgery on $K$ does not admit a hyperbolic structure.

Thurston showed that there are finitely many exceptional slopes for any knot complement, and it's now known that there are at most $10$. I'm interested in understanding what sorts of numbers $p/q$ can occur. Is anything known in general?

For example, when $K$ is the figure-eight knot, the exceptional slopes are [T, Theorem 4.7] $1/0, 0/1, 1/1, 2/1, 3/1$, and $4/1$, along with their negatives. All of these except $(1,0)$ have $1$ in the denominator. Does this pattern hold more generally?

The exception to this pattern is $(1,0)$ which is always an exceptional slope because it gives $S^3$ for any $K$, unless I have the conventions backwards and $(0,1)$ does. (Usually we consider $1/0 = \infty$ to correspond to no surgery for this reason, but that doesn't seem to be Thurston's convention.)

[T] W. Thurston, Geometry and topology of three-manifolds, http://library.msri.org/books/gt3m/

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    $\begingroup$ See Gordon’s survey for a summary. In particular, the denominator is at most 2 (see table 3.1). I’ll see if I can dig up a more recent reference (I’m not sure if the finite non lens space fillings have been classified). matwbn.icm.edu.pl/ksiazki/bcp/bcp42/bcp42111.pdf $\endgroup$
    – Ian Agol
    Commented Mar 26, 2022 at 11:50

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For an overview, see the survey of Cameron Gordon and older survey.

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Any integer can occur as the slope of a Seifert fibered Dehn filling on a knot.

Conjecture 3.3 of Gordon’s survey states that Seifert fillings must have integral slope. Thus one is left to consider reducible and toroidal fillings. Reducible fillings have integer slope. As indicated in Sam Nead’s answer, toroidal slopes are either integeral or half integral and one of Eudave-Muñoz’s examples. Hence the conjecture would give a complete characterization of exceptional Dehn filling slopes of hyperbolic knots. I’m not sure how much progress has been made on this conjecture in general. Exceptional surgeries have been classified for arborescent knots. Originally Gordon had conjectured that any Seifert surgery on a knot came from Dean’s construction, but other examples were found by Mattman-Katura-Kimihiko . Other specific classes of Seifert spaces which can arise as Dehn fillings and classes of knots have been considered, as well as collections of exceptional slopes that can occur. But that seems to be the state of the art that is known about possible exceptional slopes.

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  • $\begingroup$ If I have a hyperbolic knot $K$ in $S^3$, then $1/0$ surgery gives $S^3$, and from this table we have $\Delta(1/0, p/q) = |q| \le 2$ for any exceptional slope $p/q$. In particular this gives the denominator bound in your comment. Is that correct? $\endgroup$
    – Calvin McPhail-Snyder
    Commented Mar 27, 2022 at 20:09
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    $\begingroup$ @CalvinMcPhail-Snyder yes, exactly. $\endgroup$
    – Ian Agol
    Commented Mar 27, 2022 at 20:33
  • $\begingroup$ In this case, do we know that the hyperbolic metric on $M_{p/q}$ actually comes from that of $K$? (Here I'm thinking of this question: mathoverflow.net/questions/121774/…). I think this is equivalent to asking whether the slope $p/q$ is a geodesic in $M_{p/q}$ whenever $M_{p/q}$ is hyperbolic. $\endgroup$
    – Calvin McPhail-Snyder
    Commented Mar 28, 2022 at 14:06
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    $\begingroup$ @CalvinMcPhail-Snyder there’s two issues there: can the complete hyperbolic metric be deformed to the metric on the filling through cone metrics (or generalized Dehn fillings)? Or as you say, is the core curve a hyperbolic geodesic? The second is not known to imply the first. However the best results on both still come from the method of Hodgson-Kerckhoff. The upshot is that we don’t have very precise understanding of the exceptional slopes in this sense. $\endgroup$
    – Ian Agol
    Commented Mar 28, 2022 at 15:59
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You will perhaps be interested in the paper "Non-integral toroidal Dehn surgeries" by Gordon and Luecke. They prove:

Theorem 1.1. Let $K$ be a hyperbolic knot in $S^3$ that admits a non-integral surgery containing an incompressible torus. Then $K$ is one of the Eudave-Muñoz knots $k(l, m, n, p)$, and the surgery is the corresponding half-integral surgery.

Almost immediately before the statement they remark that the Eudave-Muñoz knots

... are the only known examples of non-trivial, non-integral, non-hyperbolic surgeries on hyperbolic knots.

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