  Trigonal PtBi$_2$ is a layered semimetal without inversion symmetry,
featuring 12 Weyl points in the vicinity of the Fermi energy. Its topological
Fermi arcs were recently shown to superconduct at low temperatures where bulk
superconductivity is absent. Here, we perform first-principles calculations to
investigate in detail the bulk and surface electronic structure of PtBi$_2$,
and obtain the spin texture as well as the momentum-dependent localization of
the arcs. Motivated by the experimentally observed recovery of inversion
symmetry under pressure or upon doping, we interpolate between the two
structures and determine the energy and momentum dependence of the Weyl nodes.
For deeper insights into the surface superconductivity of PtBi$_2$, we
construct a symmetry-adapted effective four-band model that accurately
reproduces the Weyl points of PtBi$_2$. We supplement this model with an
analysis of the symmetry-allowed pairings between the Fermi arcs, which
naturally mix spin-singlet and spin-triplet channels. Moreover, the presence of
surface-only superconductivity facilitates an intrinsic
superconductor-semimetal-superconductor Josephson junction, with the
semimetallic phase sandwiched between the two superconducting surfaces. For a
phase difference of $\pi$, zero-energy Andreev bound states develop between the
two terminations.
