The complement of the Petersen graph is the intersection graph of the 2-point subsets of {1,2,3,4,5}. Explicitly, its vertices are sets of type {i,j} with i != j among 1..5, and two vertices form an edge if they have a nonempty intersection. This delivers a 6-regular graph on 10 vertices, a picture of which is shown in the attachment.

A simplex in a graph is a clique, a set of mutually connected vertices. This makes the complement of the Petersen graph into a 3-dimensional simplicial complex with 5 tetrahedra  and with 10 triangles that are not part of a tetrahedron. The question is whether there is a (piece-wise linear) embedding of this complex into Euclidean 3-space.

The question arose from pragmatic considerations in programming an animation of lambda(5), the superextension of a 5-point set or (equivalently) the free median algebra on 5 points. As every finite median algebra, lambda(5) is (the vertex set of) a cubical complex. It is a median subalgebra of Euclidean 5-space and hence embeds in it as a cubical complex. Up to the 5 starting points, the 4-dimensional cubical complex lambda(5) turns out to be the simplex graph of the above complementary Petersen graph. See my paper (with H.-J- Bandelt), Superextensions and the Depth of Median Graphs, Jounal of Combinatorial Theory A, Vol.57, no.2 (1991), pp 187-202. It is available on RG. For a general view on (median) convexity I refer to my monograph Theory of Convex Structures, North-Holland Mathematical Library Vol.50, 1993, Elsevier Science Publishers.

A piece-wise linear embedding of the complementary Petersen graph in Euclidean 3-space would lead to an embedding of lambda(5) into Euclidean 4-space with skewed cubes, which is a good enough starting point to construct an animated presentation on screen. Otherwise, a topologically unreliable reduction would be needed at the first dimension reduction from 5 to 4.