I recently came across this paper, 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 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, 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, 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.
