The borromean knot is the solution to the following problem : How could we make hold together circles that don’t make couple? We’ll show, I hope at least, why this knot should concern psychoanalysts.

What’s to make couple ? To demonstrate it I’ll use these drawings which will be familiar to you, applied by Lacan in one of his seminars.

You quickly notice, these two drawings are different, on one hand the two triangles make a couple or a chain, on the other they are independent one from the other.

Figure 1

How does the borromean knot give an account of the structure of the psychoanalytic experience itself ?

Lacan’s proceeding has as you know consisted to notice and to point out that Freudian interpretation acts on words : open Dream’s interpretation, everyday life psychopathology, and Wit. It’s only in contemplation to that. Freud interprets from what is said of the dream’s telling or of errors, it’s a language practice.

Lacan took hold in this proceeding upon Saussure, the one he calls «the founder of modern linguistics». In fact, from the instant signifer’s dimension is isolated as such, it’s easy to show that the freudian unconscious dimension is there.

Saussure clinches in regard to language conceptions that preceed him in showing that there’s no natural link neither between signifer and signified, nor between signifer and things. On one hand he claims the «arbitrary» of the signifer organized according to a network of pure differences, one shouldn’t see labels stuck on things, that is to say a nomenclature, but a distinct order having it’s own laws and determining all effective connexion to the object.

That analytic interpretation is based on the language order is easily provable. For instance, the Ratman who loses weight not to be Dick, which in German means fat. Dick was the christian name of his rival close to his beloved and meaning «fat».

Lacan speaks out after Saussure not of the arbitrary of the link between signifer and signified but of contingency. That is to say that every language has a proper genius and organizes its proper network of preferential tracks between signifers. So where Saussure gives the model «si je la prends» et «si jel’apprends»1 to show that the cut between the signifers determines the meaning, it’s clear that for a French speaker, the word «apprendre» isn‘t far from an idea of gripping, distraint, and on the other hand in this example the sexual equivocal is obvious. As well as the word «connaître»2 of which one is aware of the biblical meaning, shows that every meaning is finally sexual.

From this closed order of the signifer, how his there meaning ? It’s the effect, says Lacan, of the substitution of a signifer to another, it’s the metaphor which produces the meaning effect. And that’s the way analytic interpretation proceeds by a meaning effect acting on the signifer equivocal .

If we report to the diagram of Saussure of the two floods of the amorphous mass of the ideas over the one of the sounds, it’s a substitution on signifer that produces an effect on the upper mass, that means in the Imaginary.

Words make things.

But if this is verified in poetical metaphor, of the analytic interpretation we are entitled to expect more, that is effects on the Real level. Analytic interpretation, is done «not to be understood but to produce waves» says Lacan. That is to say that beyond meaning, it’s by a properly literal effect that symptom is freed.

It’s the borromean knot which makes us handle this.

In fact, we have now three separate dimensions, the Symbolic, the signifer order, the Imaginary which has no natural link with the first one but where meaning is produced and the Real different from the first ones and which escapes Symbolic. Three separate dimensions that slip one in regard of the other and which are independent two by two. It’s then difficult to conceive how these separate dimensions still hold together.

Because they effectively hold together, that’s what demonstrate nomination. In fact, the name seems to assure a hold on the Real. Logicians and philosophers consider the name as a «rigid designator» to show up the privileged link with what they call «the referent».

But then they miss what makes the link between words and the Real, that means the third term, the Imaginary which they consider by mistake as the real which is in fact nothing else than what Plato called Idea.

How can three distinct things, that have no link two by two, in the same manner hold by three and only by three ?

It’s properly unconceivable, that means it can’t be thought, precisely because thought works on the pattern, on the image of the body’s bag. However psychoanalysis leads us to this consequence hard

1 «si je la prends» means «if I take her» and «si je l’apprends», «if I know if». We could give this example in the english language «If I’Il earn» and «If l’l (1) leam». 2 to know, to have to do with , to experience.

to accept : words cannot have only imaginary effects, but a language practise sometimes succeeds to transform the real for a subject. And this without any confusion of these three dimensions or a direct hold one on the other.

Only recourse to topology allows to go beyond this insufficiency of one’s own imaginary. Three independent round strings can be linked in such a way that the breaking of any string (of the three) undo the knot. (Fig. 2)

And the manipulation of this knot shows that each round can fill any order in the three (rounds) knot. (Fig 3)

Thus, in the disposal where there are two extreme circles and a middle one, it’s obvious that the two extreme circles don’t make couple. As each circle can occupy in turn these positions, there’s no couple in this knot. The knot puts within reach what is properly unimaginable.

On the same hand, it’s then possible to conceive that language game doesn’t only slip on the imaginary to bring in meaning, but that the Real itself is involved in this knotting.

But why giving to the Real, the Symbolic and the Imaginary dimensions the consistency of a string and the circle shape. There’s an oddness that strikes by it’s prominent gratuity.

Let’s consider the three consistencies of the borromean knot : The Imaginary : Why is Imaginary a hole, and why does Lacan make of the Imaginary the consistency itself of the three knot circles ?

The Imaginary is what makes body; the obsessional is particularly sensible to the fact that the body is a bag nay even a pot. And the orifices of this body are important to consider in psychoanalysis. In fact, whatever makes rim on the body may be the source of a drive and corresponds to a particular object, called «objet petit «a»» : the breast, faeces, but the look, the voice as well. But what makes hole in the Imaginary is indicated in the mirror phase.

As one knows during this mirror phase, the child, thanks to an image, establishes his body’s unity, and anticipates his motor growth. Lacan evokes a movie achieving this experiment of a child in front of the mirror: the child passes the hand before his sex hiding it in that manner in the image: this elision performance is contemporaneous to the moment the body takes hold of it’s unity.

It’s because in the image there’s a hole where stands this imaginary object, the phallus, that the body gets its unity . We must notice, this operation doesn’t depend of the child’s sex, since while the hand hides this spot, is there a penis or not ?

The important, is the absence of the imaginary phallus evoked by this elision. Of course, this operation requires the child’s entry in the language field ; the most primitive occultations game implies in fact the elementary signifer opposition, as Freud evokes it in the Fort-Da of the spool game. Out of language, absence itself is unconceivable.

The Symbolic now. Since one enters into language it’s clear one never comes out, it turns round, a signifer always sending to another signifer and there’s no need to recall here the logic begging question. In fact, logic gets its consistency from non-contradiction, even when it’s obvious that talking implies

contradiction all the time. Let’s recall the child’s image who looks for some enlightments on sex in the dictionary and who is constantly sent from a disappointing definition to another disappointing one.

This puts us on the track. The symbolic consistency is the phallus which gives meaning (sexual) to symbolic chains ; and what makes hole, is what can’t be said, it’s the primal repression, the Urverdrängt hole. The symbolic hole is linked to castration complex and to incest prohibition. Lacan makes out of the paternal metaphor the mechanism of this complex during which the Name-of-the-Father metaphorizes the phallus, the signifer of the desire.

An excellent illustration is the symbolic compact between God and his people, where the literal transformation of the name of Abram to Abraham and Saraï to Sarah, and the exigency to «cut off the fesh excrescence» seal the promise of a miraculous fecundity.

The Real to end up.

One mustn’t mix the Real evoked by Lacan and the reality which is a tamed Real. The Real is what escapes from the Symbolic and the Imaginary, it’s what stands elsewhere, what ex-sists says Lacan, what turns around the two other consistencies. If the Symbolic network determines the field of the possible, the Real is the impossible. Why is there a hole in the Real itself ?

It’s the symbolic which makes a hole in the Real. This has been well perceived by Hegel who treated the word as murder of the thing. The signifer, symbol of an absence, bores the Real. Out of language, nothing would lack to the Real. For the speaking creature the Real is bored by sexuality it’s only for the speaking creature that sexuality is a problem in the degree to make a hole in the Real.

That’s the manner the borromean knot ties the three dimensions: Real, Symbolic, Imaginary. Clinic shows then, every particular mode of binding Real, Symbolic,

Imaginary with three or with the fourth symptom string, or else in mingling them as in psychosis, when the binding couldn’t be done.

It remains to analysts to experience the relevance of this topology which implies another way of thinking.