Hyperbolic 3-manifolds with a single cusp have canonical ideal triangulations constructed by Epstein and Penner via a certain convex hull construction involving the light-cone in the hyperboloid model of hyperbolic space. (Occasionally, these “triangulations” are really cellulations more complicated cells.) When such a 3-manifold fibers over the circle, there is another type of natural triangulation, called a layered triangulation. Roughly, one starts with a certain ideal triangulation of the fiber surface, looks at the image of this triangulation under the bundle monodromy, interpolates between these by a series of Pachner moves which can then be realized geometrically by layering on tetrahedra.
When the fiber is a once-punctured torus, these two type of triangulations coincide. This was shown by Marc Lackenby using a remarkably soft and elegant argument. It’s natural to wonder whether this phenomena occurs more broadly. For instance Sakuma suggested considering the following:
Conjecture: Canonical triangulations of punctured surface bundles are always layered.
Saul Schleimer and I have discovered that this is false in general. In particular, the manifold v1348 from the SnapPea census is fibered by a once-punctured surface of genus 5 yet has a canonical triangulation which is not layered. Precisely, the canonical triangulation does not admit one of Lackenby’s taut structures so that the resulting branched surface carries something with positive weights. Note that v1349 is in fact the complement of a certain knot in the 3-sphere [CFP].
Technical details: The canonical triangulation of v1348 is in fact just the triangulation encoded in the SnapPea census (and it is a triangulation, not a celluation). It’s easy to check that it fibers using the BNS invariant (cf. [DT]), and compute the genus of the fiber from the Alexander polynomial. One then checks that there are no taut structures of this type using Marc Culler and I’s t3m Python package.