Interior of Self-Intersecting Curve

Question

Is there a natural analogue of the notion of the interior (and thus exterior) for a closed curve that is self-intersecting? For some curves I’ve come up with I can visualise it, as shown in the diagram below, and might have a definition. Red depicts the interior and blue the exterior, arrows indicate the nature of the curve as I think multiple curves with the same image could have different interiors.

I think for a closed curve $f:S^1\to\mathbb{R}^2$ you could define it as $\bigcap\left\{\textrm{range}\left(f’\right)\ \middle|\ f’:D^2\to\mathbb{R}^2\land f’|_{S^1}=f\right\}\setminus\textrm{range}\left(f\right)$. However, I don’t know if this set corresponds to the interior for non-intersecting curve, or if it is even non-empty for non-degenerate curves.

Answer 1

Here is a 14-sided polygon whose interior has been defined in several different ways (the list is not exhaustive). Note not only the variety of fillings or densities which this leads to, but also the different calculations of its area which arise in consequence. There is no "right" answer, there is only the one which best serves your current purpose.

Answer 2

There is.

The matter resolves to what a surface means, and how this moves into the next dimension.

Suppose you take a figure of 8. This is self-intersecting at the middle, yet it holds two areas of positive area (the loops). If you suppose that $S = \nabla D$, that is, surface is the gradient of density, the surface becomes an 'outwards-pointing' vector or out-vector, the intensity of which is the differences in density.

But if you suppose that the fabric ("manifold") of the surface is unaware of its aroundings, (ie the space not in the surface), then it has a density of one throughout, and is thus completely closed (ie there are no vertices since $S = \nabla D = 0$ throughout).

One possibility is that at the crossing of the surface there is a 'skew marginoid', or partition of the surface where the out-vector reverses but retains the same intensity. This allows certain polytopes, such as the seven-sided 'Thah', to keep a stable volume pattern.

Non-orientability becomes 'there exists a closed path that crosses an odd number of skew marginoids'.

This was the proposition I advanced on a mail-list to people like John Conway and Norman Johnson.