There exists an ideal triangulation, constructible by algorithm, in which the fiber of a once-punctured surface bundle over a single torus boundary is realized as an embedded spun-normal surface.
An alternative proof that 3-manifolds can be triangulated,
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Ideal Triangulations and Once-Punctured Surface Bundles
There exists an ideal triangulation, constructible by algorithm, in which the fiber of a once-punctured surface bundle over a single torus boundary is realized as an embedded spun-normal surface.