Changes

Far from being given a priori, every space is organized on the basis of cuts and can actually be considered as a cut in the space of a higher dimension. We are familiar with the subjective impact of this: The events of our lives only become history through the castration complex, which organizes our reality at the price of an imaginary cutting off of the penis. According to Freud, by introjecting a single trait of another, the subject identifies with the other (at the price of losing this person as a love object). In the single trait Lacan found the very structure of the signifier: A cut allows the lost object to fall away. He called this cut the "unary trait."

The linguist Ferdinand de Saussure insisted on the fundamentally negative, purely differential character of the signifier. Lacan formalized this property in the double loop, or "interior eight," in which the gap created by the cut is closed after a second trip around a fictional axis. The difference of the signifier from itself is indicated by the difference between the two trips around the loop (Figure 1).

If a signifier represents the subject for another signifier, then the subject would be supported by a surface whose edge would be a signifying cut. Note that the plane—the usual screen for the subject's images, figures, and dreams, that is, plans—is a surface that does not meet these conditions. The double loop cannot be drawn on a plane without showing a cut. The same is true of a sphere, a simple representation of the universe.

The Möbius strip, on the other hand, can represent this cut and symbolize the subject of the unconscious. Since a Möbius strip only has one surface, it is possible to pass from one side to the other without crossing over any edge—an apt representation of the return of the repressed. The Möbius strip also has certain other peculiarities. A cut that runs one-third from the edge and parallel to the edge divides the strip into a two-sided strip linked to what remains of the original Möbius strip. But if this cut is made in the center, it does not divide the Möbius strip in two. Instead, the entire strip is transformed into a strip with two sides. This characteristic illustrates the equivalence between the Möbius strip (the subject) and the medial cut that transforms it, and also provides a model of how interpretation functions. Interpretation does not abolish the unconscious. On the contrary, it makes the unconscious real for the subject by its transformed appearance as another (an Other) surface (figure 2).

Lacan made different uses of the torus. By drawing Venn diagrams, traditionally used to illustrate basic logical operations, on the surface of the torus, he demonstrated the extent to which our thinking depends upon the plane surface, and he also provided another possible basis for the logic of the unconscious (Figure 3).

For every torus, there is a complementary torus, and the empty spaces of the two are the inverse of each other. Lacan made this structure of complementary toruses the support of the neurotic illusion that makes the demand of the Other the object of subject's desire and, conversely, makes the desire of the Other the object of subject's demand. This structure also arises from the fact that on a torus, the signifying cut (the double loop) does not detach any fragment. Neurotic subjects, insofar as they give in to neurosis, insofar as they are "in the torus," are not organized around their own castration, but instead excuse themselves by substituting the Other's demand for the object of their fantasy (figure 6).

The cross-cap, or more precisely, the projective plane, can represent the subject of desire in relation to the lost object. A double loop drawn on its surface in effect divides this single-sided surface into two heterogeneous parts: a Möbius strip representing the subject and a disk representing object a, the cause of desire. The disk is centered on a point that is related to the irreducible singularity of this surface, which Lacan identified with the phallus. Unlike the representation of the subject produced on the torus, here a single cut, which symbolizes castration, produces both the subject and the object in its divisions (figure 7).

Introduced by Lacan in 1973, the Borromean knot is the solution to a problem perceivable only in Lacanian theory but having extremely practical clinical applications. The problem is: How are the three registers posited as making up subjectivity—the real (R), the symbolic (S), and the imaginary (I)—held together?

By using knots, Lacan was able to reveal his ongoing research without hiding its uncertainties. The value of the knots, which resist imaginary representation, is that they advance research that is not mere speculation and that they can grasp—at the cost of abandoning a grand synthesis—a few "bits of the real" (Lacan, 1976-1977, session of March 16, 1976). Even though he knew something about topology as practiced by mathematicians, Lacan advised his students "to use it stupidly" (Lacan, 1974-1975, session of December 17, 1974) as a remedy for our imaginary simplemindedness. He also recommended manually working with the knots by cutting surfaces and tying knots. Finally, for Lacan, topology had not only heuristic value but also valuable implications for psychoanalytic practice.

Lacan criticises these models for not being topological enough.

It is thus the task of Lacan's topological models "to forbid imaginary capture."<ref>E, 333</ref>

Unlike intuitive images, in which "perception eclipses structure", in Lacan's topology "there is no occultation of the symbolic."<ref>E, 333</ref>

Lacan argues that topology is not simply a metaphorical way of expressing the concept of structure; it is structure itself.<ref>Lacan, 1973b</ref>

While [[schema L] and the other schemata which are produced in the 1950s can be seen as Lacan's first incursion into topology, topological forms only come into prominence when, in the 1960s, he turns his attention to the figures of the [[torus]], the moebius strip, Klein's bottle, and the cross-cap.<ref>see Lacan, 1961-2</ref>

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