  In this work, we undertake a perturbative analysis of the topological
non-Abelian Chern-Simons-Wong model with the aim to explicitly construct the
second-order on-shell action. The resulting action is a topological quantity
depending solely on closed curves, so it correspond to an analytical expression
of a link invariant. Additionally, we construct an Abelian model that
reproduces the same second-order on-shell action as its non-Abelian
Chern-Simons-Wong counterpart so it functions as an intermediate model,
featuring Abelian fields generated by currents supported on closed paths. By
geometrically analyzing each term, we demonstrate that this topological
invariant effectively detects the knotting of a four-component link.
