Among all one-arm labyrinths with up to and including 7 circuits, there are no two uninteresting labyrinths complementary to each other. The reason for this is that in such labyrinths the pathway always must enter the labyrinth on the outermost circuit or reach the center from the innermost circuit (see related posts, below). However, there exist uninteresting labyrinths with more than 7 circuits in which this is not the case.
In order to show this, I begin with the example of the 11-circuit Cakra-Vyuh labyrinth (see related posts). Figure 1 shows this labyrinth and the pattern of it.
Figure 1. The 11-circuit Cakra Vyuh Labyrinth
As can be seen, the pathway enters the labyrinth on the first circuit and reaches the center from the innermost circuit. So, the outer- and innermost circuits can simply be cut-off (grey lines in the right image). This then results in a labyrinth with 9 circuits, in which the pathway does not enter on the outermost circuit and doesn’t either reach the center from the innermost circuit. The pattern of this labyrinth is shown in figure 2.
Figure 2.The Pattern of the Uninteresting Labyrinth with 9 Circuits
Because we removed the grey circuits, the course of the pathway in the remaining pattern is from top right to bottom left. If we want to show the pattern in the usual form, we have to mirror it horizontally. This does not affect the pattern itself nor the labyrinth related to it, except for the labyrinth changing its rotational direction (see related posts).
Even though the pathway of this labyrinth enters on the 3rd circuit and reaches the center from the 7th circuit, this is an uninteresting labyrinth. This, because it is made up of two elements of the type Knossos on circuits 1 – 3 and 7 – 9 (indicated with brackets in the right image) and three internal trivial cirucits 4, 5, 6 between them (indicated with dashes). Although this labyrinth is uninteresting, it is self-dual.
Parenthesis: This labyrinth has similarities with the well known basic type (former: Cretan type) labyrinth. However, the basic type is a very interesting (that is interesting and self-dual) labyrinth.
Figure 3. The Pattern of the Basic Type Labyrinth
As shown in figure 3, this is also made-up of two elements of the type Knossos. However, between these there is only one circuit. And this is by no means trivial as it is needed to connect the two elements. But adding further circuits in the shape of a serpentine will result in an uninteresting labyrinth.
Let us get back to the uninteresting labyrinth with 9 circuits. How does the complementary labyrinth look like? Is it may be also an uninteresting labyrinth?
Figure 4. The Two Complementary Labyrinths with 9 Circuits
In order to generate the complementary, we mirror the original labyrinth vertically and let the connections with the environment and the center uninterrupted. Then the pathway enters on the 7th circuit and reaches the center from the 3rd circuit. The three trivial internal circuits are still recognizable. However, they are enclosed by the axial pieces of the pathway that lead into the labyrinth and to the center. So they are nested one level deeper. Therefore, this is no more an uninteresting, but an interesting, and, as it is self-dual, a very interesting labyirnth.
Thus it seems, that also in larger one-arm labyrinths there are no pairs of uninteresting labyirnths that are complementary to each other.