  The topological charge of the spin texture in a quantum dot with spin-orbit
couplings is shown analytically here to be stable against the ellipticity of
the dot. It is directly tunable by a single magnetic field and is related to
the \textit{sign} of the Land\'e $g$ factor. In a quantum-dot helium, the
overall winding number could have different property from that of the
single-electron case (quantum-dot hydrogen), since tuning the number of
electron affects the winding number by the Coulomb interaction and the $z$
component angular momentum $\langle L^{}_z \rangle$. The density profile and
the spin texture influence each other when the Coulomb interaction is present.
When $\langle L^{}_z \rangle$ is biased away from an integer by the spin-orbit
couplings, the rotational symmetry is broken which induces strong density
deformation. The sign of the topological charge may also be reversed with
increasing magnetic field. These findings are of major significance since the
applied magnetic field alone now provides a direct route to control the
topological properties of quantum dots.
