An edge of a simplicial complex is said to be *contractible* if it may be contracted to a vertex without modifying the topological type of the underlying manifold. Otherwise, the edge is *noncontractible*. 

Not every edge of a simplicial complex is contractible. For example, on a [triangular bipyramid](http://upload.wikimedia.org/wikipedia/commons/a/a5/Triangular_dipyramid.png), the edges gluing the two pyramids together are noncontractible, while all other edges are contractible (yielding a simplex). On a $d$-simplex, no edge is contractible.

[Dey, Edelsbrunner, Guha, and Nekyahev](http://www.cs.duke.edu/~edels/Papers/1999-J-03-TopologyPreservingContraction.pdf) provide exact conditions for when an edge of a 2- or 3-manifold is contractible, noting that an edge $ab$ between vertices $a$ and $b$ is contractible iff  $link(ab) = link(a) \cap link(b)$.  I have found little in the literature about conditions for the contractibility of edges on higher-dimensional objects.

My questions concern the existence of contractible edges on polytopal $d$-spheres, ie, simplicial polytopes.  

1) Does every simplicial polytope other than the simplex have a contractible edge?

2) Does contraction of an edge of a simplicial polytope always yield another polytope?

3) Are these questions any easier to answer if we restrict the question to 4-polytopes?