**6. Topology**

Strangely, the topological deconstruction of the knot and its movement on an orbit produces the same design. Foremost it is the topology that transforms appearances and ties together many seemingly different images.

Renaissance painters that mastered perspective were called geometers. Today, the geometry that permits exploration of new areas is topology. It is linked to DNA, quantum physics and string theory. One finds it again with the catastrophe theory, and even in psychology with Lacan, who borrowed the Borromean knot as a model for the psyche. If perspective rendered the image of the gods more real and approachable as figures of the invisible, topology reveals the invisible in concrete figures .

Topology is geometry of networks, knots and relationships: the “geometry with a twist”. Topology is in fact the basis of a science of links that Giordano Bruno said constituted magic. It is because it is based on topology that our work reveals meaningful links between places and concepts.^{1} It is topology that guides the mental journey from the cat’s cradles towards the Khora.

Because it is very difficult to imagine the relationship between the interior and exterior in current life, we therefore need models. It is what topology offers. As the knot becomes unfolded and divided we enter into the world of in-between. The model of this world also helps to understand and chart what is between the intelligible and the sensible, between the visible and the invisible

With topology there is a geometry corresponding to what Merleau-Ponty proposes to support a phenomenological approach: “Replace the notions of concept, idea, mind, and representation with the notions of dimensions, articulation, level, hinges, pivots, configuration (…)”.^{2}

Let’s add that this topology as it is linked to symbols becomes a mathesis, such as Deleuze defines it: * “Mathesis is neither a science, nor a philosophy. It is something else: a knowledge of life... It is situated on a plan where the life of knowledge identifies itself with the knowledge of life, where it is simply an awareness of life”.*

By painting knots and their interlacings, one produces a modern version of many ancient arts, most notably Celtic art. One wonders about the relationship between this art and the religion (that which links) or about the belief system of the Celts. In fact, each different kind of knot might represent a different kind of relationship. Modern topology could help to explore this ancient system.

The construction of the labyrinth introduces a paradox. The fact that the knot is open and doubled, gives way to a fourth dimension, which no longer appears when mapping the knot around an orbit. It then replace by time. (see 2 The Construction of the Labyrinth). Like the invisible, the fourth dimension is no longer something added. Rather, it is like the discovery of an inner and more piercing gaze, or like the perception of an overtone.

Topology also reveals that the knot (our symbol of the whole) is contained in a torus. (fig. 58). (A torus with three holes, in the case of the knot with seven crossings which produces the Cretan labyrinth) According to this model, we would be living mentally in a circular and pierced spaced as within the interior of a tire. We would always circulate around an unknown void beyond our comprehension. We would be ignorant, like the dwellers of Plato’s cave. One can then wonder if following the motion and spin of the knot, the torus can reverse itself or close on itself and if the labyrinth can give access to these holes which pierce our world.

n°58 Modèle du noeud inscrit dans un tore

^{1}Actually, in modern times, topology started with the problem posed by the Koenigsberg bridges.

^{2}Merleau-Ponty ‘The Visible and the Invisible