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Could anyone show me or refer me to a link where the volume of a hyperbolic knot, say, the figure-8 knot, is computed (well, in fact estimated) explicitly and not only having the procedures outlined?

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You can look at Thurston's notes where he explains (in Chapter 1 and Section 3.1) that the figure-8 knot complement can be obtained by gluing two ideal tetrahedra (with dihedral angles $\pi/3$) together. One can compute the volume of an ideal tetrahedron in terms of its dihedral angles using a formula given in Theorem 7.2.1.

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