  $\alpha$-Sn is well known as a typical Luttinger semimetal with a quadratic
band touching at the $\Gamma$ point. Based on the effective $k\cdot p$ analysis
as well as first-principles calculations, we demonstrate that multiple
topological phases with a rich diagram, including topological insulator, Dirac
semimetal, and Weyl semimetal phases, can be induced and engineered in
$\alpha$-Sn by external strains, magnetic fields, and circularly polarized
light (CPL). Intriguingly, not only the conventional type-I Weyl nodes, but
also type-II Weyl nodes and double-Weyl nodes can be generated directly from
the quadratic semimetal by applying a magnetic field or CPL. Our results apply
equally well to other Luttinger semimetals with similar crystal and electronic
structures, and thus open an avenue for realizing and engineering multiple
topological phases on a versatile platform.
