  We demonstrate a topological classification of vortices in three dimensional
time-reversal invariant topological superconductors based on superconducting
Dirac semimetals with an s-wave superconducting order parameter by means of a
pair of numbers $(N_\Phi,N)$, accounting how many units $N_\Phi$ of magnetic
fluxes $hc/4e$ and how many $N$ chiral Majorana modes the vortex carries. From
these quantities, we introduce a topological invariant which further classifies
the properties of such vortices under linking processes. While such processes
are known to be related to instanton processes in a field theoretic
description, we demonstrate here that they are, in fact, also equivalent to the
fractional Josephson effect on junctions based at the edges of quantum spin
Hall systems. This allows one to consider microscopically the effects of
interactions in the linking problem. We therefore demonstrate that associated
to links between vortices, one has the exchange of quasi-particles, either
Majorana zero-modes or $e/2$ quasi-particles, which allows for a topological
classification of vortices in these systems, seen to be $\mathbb{Z}_8$
classified. While $N_\Phi$ and $N$ are shown to be both even or odd in the
weakly-interacting limit, in the strongly interacting scenario one loosens this
constraint. In this case, one may have further fractionalization possibilities
for the vortices, whose excitations are described by $SO(3)_3$-like conformal
field theories with quasi-particle exchanges of more exotic types.
