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Some computations show that the Weeks manifold and the 5.2 knot complement have the same trace field (which is $\mathbb{Q}[x]/(x^3-x+1)$) and the (hyperbolic) volume of the second is 3 times the volume of the first.

My question is: is there a geometric explanation?

They cannot be commensurable, as one is compact and the other isn't. I also have an algebraic explanation: the Bloch group of the trace field has rank 1, but I would prefer a geometric one, explaining the factor 3.

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One explanation is to look at the "spun triangulation" SnapPy gives to the Weeks manifold. This is composed of two ideal tetrahedra with volumes ~0.83828404504 and ~0.10442331774 adding to the familiar ~0.94270736278. On the other hand, the triangulation SnapPy gives to the complement of the $5_2$ knot consists of three ideal tetrahedra, all with volume ~0.94270736278.

We can check that all of these various volumes agree by solving the gluing equations (ensuring (a) that there is no rotation or shearing about edges and (b) that the cusps are correctly (in)complete).

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    $\begingroup$ Thanks for this very precise answer although I find it somewhat disappointing: it looks like a bare coincidence. Don't you think there could exist a better explanation? $\endgroup$
    – Julien Marché
    Commented Jan 12 at 22:16
  • $\begingroup$ I really don't know. There are many many "volume coincidences" to be found in the SnapPy censuses. Some of these are explained by commensurability, some are explained by more-or-less subtle mutations, and some are just mysterious. Volume coincidences between cusped and closed manifolds are of the latter type. In my limited experience, there is always a fairly short scissors congruence between spun tetrahedra in the closed manifold and ideal tetrahedra in the cusped manifold. (And If I can't find one, I ask Craig Hodgson.) $\endgroup$
    – Sam Nead
    Commented Jan 13 at 7:36

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