6.4.1 Disambiguation rule for 4-hit faces.

Let us look at the situation in 2 dimensions. Let A be the second largest value of $\delta$ on the face F, that is, the value at which the diagonally opposite vertices of F become High, and let B<A be the value at which a third vertex becomes High. From the above reasoning we can conclude that we must select a value c, such that A < c < B, so that above c the Lows are adjacent, and below c the diagonally opposite Highs are adjacent. We call c the disambiguation value. If c is chosen so that -c is also the disambiguation value for $-\delta$, then this gives the desired symmetry property. This is because if we choose the Highs of $\delta$ as adjacent for $\tau < c$ it means that we make the same two vertices (as Lows of $-\delta$) adjacent for $-\tau > -c$ (and vice versa). Thus the topology of $\bf {X}_{-f}^{> - \tau}$ and $\bf {X}_{f}^{< \tau}$ will be the same.

We call this disambiguation rule Knit High Above, Knit Low Below, and it is illustrated in the series of figures 6 through 9. It is easy to see that the complementary rule of knit Low for values above the disambiguation value c, and then knit High below the value c, is inconsistent in two dimensions, as above c the Highs are path connected through the face, and so must remain so for all thresholds $\tau < c$.