3.3 Why edge-connectivity is insufficient.

Definition 6   We call the interpolated boundary intersection points of $B (\bf {X}_{f}^{\geq \tau})$ with cube edges, ``hit points''.

Definition 7   When a hypercube face contains diagonally opposite Highs and diagonally opposite Lows we call it a 4-hit face, since by the above assumptions there is an intersection point with $B (\bf {X}_{f}^{\geq \tau})$ on each edge (see figure 4).

Definition 8   Choosing the diagonally opposed Highs in a 4-hit face as adjacent means that we will regard then as path connected through the cube face F, that is part of the same component of $F \cap \bf {X}_{f}^{\geq \tau}$. In this case a pair of hits on edges that share a common Low vertex will be connected by a boundary curve within the face. Thus we call this choice ``Knit Low''. Similarly, the choice that makes the Highs nonadjacent (and thus the Lows adjacent) is termed ``Knit High'' (see the squares in figures 4 and 5).

These are the only two choices for face F, since if there is a path $\pi \subset \bf {X}_{f}^{\geq \tau} \cap F$ between the two Highs then there cannot be a path between between the two Lows in $\bf {X}_{f}^{< \tau} \cap F$, as it would have to cross $\pi$ (and conversely). In two dimensions these will be our only two choices. As we shall see we will make a similar choice for 3 and higher dimensions. In this case we will regard them as adjacent if we determine that they are path connected through the interior of a cube sharing face F, however, as we shall see when we discuss critical 4-hit faces, no topological generality will be lost if we assume the path is through F

We now explain the sense in which edge-connectivity is inconsistent.

Proposition 3.1   If f interpolates $\delta$ then $\bf {X}_{-f}^{> - \tau} = \bf {X}_{f}^{< \tau}$.

Proof: If f interpolates $\delta$ then -f interpolates $-\delta$ since $f(p) = \delta(p)$ implies that $-f(p) = -\delta(p)$. Since $f(p) < \tau$ implies $-f(p) > -\tau$, then $\bf {X}_{-f}^{> - \tau} = \bf {X}_{f}^{< \tau}$.

Proposition 3.2   Using only edge adjacency to determine path connectivity of Highs can give $\bf {X}_{-f}^{> - \tau} \neq \bf {X}_{f}^{< \tau}$

Proof:

If we reverse the signs of all the readings, the High vertices of -f are the Low vertices of f (and vice versa). The Low vertices of the 4-hit face F are adjacent in $\bf {X}_{f}^{> \tau}$. Thus making these same two vertices non-adjacent as Highs of $\bf {X}_{-f}^{> - \tau}$ yields $\bf {X}_{-f}^{> - \tau} \neq \bf {X}_{f}^{< \tau}$ (compare figure 4 with figure 5).