The Jordan curve theorem states that if $f:S^1\to \mathbb R^2$ is an injective continuous function then $\mathbb R^2\setminus \text{image}(f)$ has two connected components. I want to discuss an approach to proving this theorem which is as follows.
Let us instead try to prove the following. Let $f:S^1\to S^2$ be an injective continuous function. Then $S^2\setminus \text{image}(f)$ has two connected components.
We have $f$ depicted in the following figure in pink.
Now thinking of $S^2$ as the surface of Earth, and thinking of the iamge of $f$ as a massive string.
We endow the image of $f$ with a uniform positive charge density so that the string starts moving (since it is repelling itself). To keep the string from flying off the planet let us also switch on gravity.
This should homotope the image of $f$ into a perfect circle. Now one would need to prove that the number of components that any intermediate curve (during gravity assisted homotopy) partitions the sphere into does not change with time, whence the Jordan curve theorem (for the sphere) can be proved since the number of components induced by the final curve is clearly $2$.
I guess one should be able to define an energy function of the system and show that the system has the tendency to go towards the lowest energy configuraton. This configuration shuld be the one where the pairwsie distances of the charges is the greatest, which happens when the string is a perfect circle. Of course, there would be many technical problems along the way.
I do not know if a proof of the Jordan curve theorem already exists along these lines. If not, then I think a proper execution of this idea should make for a publication.
EDIT: Thanks to copperhat for pointing out a flaw in the previous post where there was no mention of an electric field.
This proposal appears to be similar to the idea of the gradient flow of a knot energy, see
Simon Blatt, The gradient flow of O’Hara’s knot energies, Math. Ann. 370, No. 3-4, 993-1061 (2018). ZBL1398.53071.
(O'Hara's knot energies are certain regularizations of the energy given by the electrostatic potential.)
In his paper Blatt proves that, given a smooth knot in $S^3=R^3\cup \{\infty\}$, the gradient flow converges to critical points of the energy functional. If I understand this correctly, in the case of an unknot, the only critical points are round circles. (The trouble is that there are so many different versions of the knot energy that I cannot keep track which of these satisfy this uniqueness property.) In particular, this applies to knots contained in round spheres $S^2\subset S^3$. In view of the reflection symmetry, if you apply the gradient flow to a knot contained in a round sphere, the flow will be contained in that sphere.
This is all interesting for its own sake and for the sake of potential applications of the knot theory (say, to study DNA and protein folding), but should not be used in order to prove Jordan Curve theorem for smooth curves, since the latter has a very easy proof using transversal intersections with rays. None of these will work for topological Jordan curves.