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Say we first have two curves, $C_1$ and $C_2$ which are knotted together. Let $C_2'$ be a continuous deformation of $C_2$ such that $C_2$ does not cross $C_1$ as it is deformed into $C_2'$. How could I prove that the two linking integrals have the same value? What I had in mind so far was to show that $\mathrm{Lk}(C1,C2)-\mathrm{Lk}(C1,C2')=0$, hopefully with the use of Green's/Stokes' Theorem in some way and without any physics.

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