Hyperbolic manifolds without positive spun triangulations

Using a result of Choi, we provide the first examples of pairs consisting of a closed hyperbolic three-manifold and a simple closed geodesic, such that there is no positive spun ideal triangulation for the manifold, spun about the chosen geodesic.

In our first two examples, the closed manifold is the third manifold in the SnapPy census, also known as Vol3, and the geodesics are its systole and second systole.

This provides evidence for the conjecture that Vol3 has no positive spun ideal triangulation for any choice of geodesic.