  Via a computer search, Altshuler and Steinberg found that there are 1296 +1
combinatorial 3-manifolds on nine vertices, of which only one is non-sphere.
This exceptional 3-manifold $K^{3}_{9}$ triangulates the twisted $S^{2}$-bundle
over $S^{1}$. It was first constructed by Walkup. In this paper, we present a
computer-free proof of the uniqueness of this non-sphere combinatorial
3-manifold. As opposed to the computer-generated proof, ours does not require
wading through all the 9-vertex 3-spheres. As a preliminary result, we also
show that any 9-vertex combinatorial 3-manifold is equivalent by proper
bistellar moves to a 9-vertex neighbourly 3-manifold.
