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In Example 1.4 of his book Invariants of Homology 3-Spheres, Saveliev noted that twist knots $K_n$ of type $(2n+2)_1$ are all hyperbolic. Here, $K_1$ is the figure-eight knot $4_1$ and $K_2$ is the stevedore knot $6_1$, see the following figure

enter image description here

Actually, this seems a well-known fact to experts but I couldn't find an explicit reference. Is there an easy way to see this?

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    $\begingroup$ Check out Neumann-Reid’s Arithmetic of 3-manifolds. They classify the hyperbolic Dehn fillings on one component of the whitehead link, which includes all the twist knots. math.columbia.edu/~neumann/preprints/nrarith.pdf $\endgroup$ – Ian Agol Jul 5 '20 at 15:42
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    $\begingroup$ @Maxim: what is your preferred way to "see" a manifold as hyperbolic? On the passive side you could argue the manifold isn't Seifert-fibred and has no incompressible tori. On the more active side you could construct the hyperbolic structure by hand. Viewing them as fillings of the Whitehead link, as Ian suggests, gets you started down that road as the Whitehead link has a triangulation you can use. $\endgroup$ – Ryan Budney Jul 5 '20 at 21:11
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    $\begingroup$ Not exactly what you're after, but this fantastic paper, math.columbia.edu/~jb/new-twist.pdf, by Joan Birman and Ilya Kofman studies twisted torus knots, and iterated twistings of torus knots. They could not determine hyperbolicity, but they did manage to give an algorithm, and its complexity, to decide if twisting gives a torus knot or not. So from there you could, as Ryan Budney suggests, proceed to rule out incompressible tori. The fact that they couldn't determine hyperbolicity perhaps suggests there isn't an easy way to see hyperbolicity for general twist knots. $\endgroup$ – guest Jul 5 '20 at 22:22
  • $\begingroup$ Thanks all for the great comments. Actually, I'm interested in constructing the hyperbolic structure by hand. See for example pg. 17 in math.csi.cuny.edu/~abhijit/talks/knot-geometry-h.pdf It was done by the figure-eight knot. I just thought that it seems to be generalized to all twist knots. $\endgroup$ – user160180 Jul 6 '20 at 8:15

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