Triangulations of the `magic manifold' and families of census knots

We describe five ideal triangulations of the 3-cusped hyperbolic `magic manifold' that are each compatible with well-established techniques for triangulating Dehn fillings. Using these techniques, we construct low-complexity triangulations for all partial fillings of the magic manifold, and in particular, recover minimal triangulations for 229 of the hyperbolic census knots. Along the way, these census knots are sorted into 42 families related by twisting that can be extended indefinitely, with each member of each infinite family inheriting an upper bound on its triangulation complexity. These triangulations are conjectured to be minimal for all 42 families.