6.4.3 Critical 4-hit faces defined.

Definition 10   We define a 4-hit face as critical, if for all $\tau > c$, 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.

Now at value c the two diagonally opposite Highs become part of the same path connected component of Highs within some hypercube sharing the face F. If there is no path bewteen them passing through at least one other High vertex within the cubes sharing F at this value then, with respect to the cubes sharing F, two separate components have merged. For example in 3 dimensions, let F be shared by cubes C₁ and C₂ and let the diagonally opposite Highs be H₁ and H₂. Suppose at the value c there is no path of adjacent High vertices that travels from H₁ to the opposite face of C₁, and back to H₂ and there is no path of adjacent Highs from H₁ to the opposite face of C₂ to H₂, then we have formed a handle between two locally disconnected regions. Globally either two separate surfaces have merged or a surface has increased in genus. Even if the actual path by which H₁ and H₂ become connected is not in the face F, but say, through the interior of C₁, no topological generality is lost by moving the path to the face F. The important point is that any path of High vertices connecting H₁ and H₂ within the union of C₁ and C₂ must pass directly from H₁ to H₂. This will mean (see axioms below) that a distinct handle or bridge between the components containing H₁ and H₂ has been formed in the union of C₁ and C₂. This will indicate a topological change in the boundary components of our level sets.

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.

Even if H₁ and H₂ were already part of the interior of the same manifold M globally at the disambiguation value, we can see that any path from H₁ to H₂ can be surrounded by a loop through the interior of the union of C₁ and C₂, and this loop lies completely exterior to M.

Some may be concerned that the critical 4-hit faces introduce too many saddle point topological transitions in $B (\bf {X}_{f}^{\geq \tau})$, but we will address this issue in the section on algorithmic issues.

We are now ready to give our axiomatic presentation of the preceding discussion.