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The question of the dependence of the topological charge $q$ of a gaugedSkyrmion, on the gauge field, is studied quantitatively. Two examples, bothgauged with $SO(2)$ are studied and contrasted: i) The $O(3)$ model in $2+1$dimensions, and ii) The $O(4)$ model in $3+1$ dimensions. In case i), where the(usual) Chern-Simons (CS) term is present, the value of $q$ changes sign, goingthrough zero. This evolution is tracked by a parameter characterising thesolutions in the given theory. In case ii), in which dimensions no CS densityis available, the evolution of $q$ is not observed.

On the topological charge of $SO(2)$ gauged Skyrmions in $2+1$ and $3+1$ dimensions