I recently came across [this paper](https://doi.org/10.1016/S0040-9383(01)00034-9), which showed that any compact smooth manifold is diffeomorphic to a connected component of the moduli space of a planar linkage.
Briefly, if we have an undirected graph $G = (V, E)$ together with a set of positive edge weights $w : E \to \mathbb{R}$, a *configuration* is a map $\phi : V \to \mathbb{R}^2$  such that, for all edges $e = \{v_1, v_2\}$ in $G$,

$$\|\phi(v_1) - \phi(v_2)\|^2 = w(e)^2.$$

The moduli space is the set of all configurations $\phi$.
See for example the [Peaucellier-Lipkin](https://en.wikipedia.org/wiki/Peaucellier%E2%80%93Lipkin_linkage) linkage, which converts straight-line motion into rotary motion and vice versa.

I had also at one point heard about the existence of [exotic spheres](https://en.wikipedia.org/wiki/Exotic_sphere), smooth manifolds that are homeomorphic but not diffeomorphic to the standard differential structure on the sphere.
That page also gives the example of [Brieskorn spheres](https://en.wikipedia.org/wiki/Exotic_sphere#Brieskorn_spheres), which are a concrete realization of an exotic sphere as an algebraic set.

My question is: **can someone explicitly compute a planar linkage that realizes some exotic sphere?**
By this I mean write down explicitly the adjacency matrix of the graph and the edge weights.
I'm not picky -- any exotic sphere will do.
My algebraic geometry kung fu is virtually nonexistent so I don't think I can figure this out myself.
I'm interested in this because I think it would be a fun party trick.
Maybe someone who teaches differential topology would find it to be a useful classroom example too.