,
the disambiguation
value,
the diagonally opposite Highs are not path connected (by adjacent Highs,
see definition 11 below) within any n-dimensional
hypercube that shares the 4-hit face.
Note that in 2 dimensions all 4-hit faces are critical. We see in figure 33 through figure 35, that no topological change occurs at the disambiguation value, while in figure 36 through figure 39 a distinct handle is formed in the cubes. In each example, the 4-hit face is the shared face between two vertically stacked cubes. The top cube face of the upper cube contains only Low vertices in these examples, and thus lies completely outside any level set component. Note that the view of the shared 4-hit face may be considered an overhead view through a transparent cube face, as it makes no topological difference whether the connection between the High vertices is actually in the face, or through, say, the interior of the cube below this face. In figure 33 through figure 35 we see that the Highs of the shared 4-hit face are already connected through the bottom cube, above the disambiguation value. We show (up to topological equivalence, see the axioms below) representative level set components surrounding the bottom cube. In figure 36 through figure 39, the Highs are not connected through the bottom cube or the top cube above the disambiguation value, and in each case a distinct connection is formed at the disambiguation value. Thus the shared face is not a critical 4-hit face in figure 33 through figure 35, but is a critical 4-hit face in figure 36 through figure 39.
We are now ready to give our axiomatic presentation of the preceding discussion.