# Talk:Borromean knot

The fundamental structure in Lacanian psychoanalysis is a tripartite confluence of what Lacan called the real, the imaginary, and the symbolic orders.1 I will define each of these in turn shortly, but first it is important to conceive of their interrelationship as "the fundamental classification system around which all [Lacan’s] theorising turns" (Evans 132). The intersection of the RSI constitutes the whole of the mental life of humans, whether in a cumulative way or in the various effects it produces – "together they cover the whole field of psychoanalysis" (Evans 132). Each of the orders not only constitutes a particular aspect of the mental life of the mature human, but also corresponds roughly to stages in the development of the infant human as it approaches maturity. Nonetheless, while it is tempting to think of the orders as stages through which the individual moves, we must resist this temptation and retain their purity as orders or registers in which, through which, and by which the individual is determined: "The symbolic, the imaginary and the real are not mental forces, personifiable on the model-builder’s inner stage, but orders each of which serves to position the individual within a force-field that traverses him" (Bowie 91). This insistence on the existence of RSI as orders or forces that traverse the individual allows us to comprehend that though together they comprise a structure, that structure is far from static. Rather, the various orders contained in the RSI configuration constantly act on each other, defining each other and themselves in contradistinction to one another. They are simultaneously mutually interdependent for their definition and utterly incommensurable. "The symbolic, the imaginary and the real pressurize each other continuously and have their short-term truces, but they do not allow any embracing programme for synthesis to emerge inside or outside the analytic encounter. The three orders together comprise a complex topological space in which the characteristic disorderly motions of the human mind can be plotted" (Bowie 98-99). Like a perpetually stymied dialectic, the three orders define themselves in purely negative relationships to each other yet never come to a point of Aufhebung at which each is subsumed by the others to produce a clear, pure, and non-pathological synthesis. The interaction of these three orders produces the analysable human subject even as their encroachment upon each other produces a variety of more or less serious disruptions in that subject.

Perhaps the easiest way to conceive of the interaction of the RSI is to employ Lacan’s model of the Borromean knot.

Figure 1 - a 2D representation Figure 2 - a 3D representation.2

The Borromean knot is a topological conceptualisation of the RSI in which each order is depicted as a circle that links each of the other orders. It is "a way of illustrating the interdependence of the three orders of the real, the symbolic and the imaginary, as a way of exploring what it is that these three orders have in common" (Evans 19-20). Its chief value lies in the fact that it "is formed from two separate links joined to each other by a third, and in such a way that if any one of the links is severed the whole thing falls apart" (Bowie 194). That is, each of the orders is fundamental to the whole in such a way that the separation of any one would automatically result in the collapse of the entire nexus, with catastrophic results for the individual constituted and traversed by it: "‘each term is sustained only in its topological relation with the others’" (Lacan S11 89, qtd. in Evans ix).

This feature of the Borromean knot generates one of the keys to understanding Lacan’s quasi-dialectical conception of human being: "that which is excluded from sense-making is that which makes sense hang together; no two agents or qualities or postulates can be coupled or contrasted without a mediating third. Everything that exists ex-sists – has its being in relation to that which lies outside it – and dichotomies and complementarities are no exception to the rule" (Bowie 194). The basic thrust of Bowie’s explication of the RSI is that each order defines itself in a negative relation to the other two orders, generating its positive attributes primarily by excluding some aspect of one or both of the other orders. Thus the symbolic is that which utterly excludes the real and which dissolves the imaginary. It relies for its internal consistency on the constant and unwavering exclusion of the other two orders, producing a definitional logic of (b)orders and their impossible/inevitable transgressions: this is the meaning of Bowie’s claim that "Everything that exists ex-sists – has its being in relation to that which lies outside it." The location of some incommensurable other against which it can set itself is the fundamental condition for each of the orders to maintain not only its consistency but its very existence. This characterisation points to the fundamental role played by antagonism and aggressivity in the Borromean Knot as each of the orders fights for its supremacy by attempting to annihilate the conditions of its own existence (i.e. the other two orders). The tensions, pressures, and cross-order "cuts" produced by this conflict constitute both the central phenomenon with which we are here concerned, the subject, and the various discontents that plague him or her.

## Contents

## def

Borromean knot (noeud borromÈen)

References to knots can be found in Lacan's work as early as the 1950s (e.g. E, 281), but it is not until the early 1970s that Lacan begins to examine knots from the point of view of their topological properties. The study of knot theory marks an important development in Lacan'S TOPOLOGY; from the study of surfaces (the moebius strip, the torus, etc.) Lacan moves to the much more complex area of the topology of knots. Topology is increasingly seen as a radically non-metaphorical way of exploring the symbolic order and its interactions with the real and the imaginary; rather than simply representing structure, topology is that structure. In this late period of his work, one kind of knot comes to interest Lacan more than any other: the Borromean knot. The Borromean knot (shown in Figure 1), so called because the figure is found on the coat of arms of the Borromeo family, is a group of three rings which are linked in such a way that if any one of them is severed, all three become separated (S20, 112). Strictly speaking, it would be more appropriate to refer to this figure as a chain rather than a knot, since it involves the interconnection of several different threads, whereas a knot is formed by a single thread. Although a minimum of three threads or rings are required to form a Borromean chain, there is no maximum number; the chain may be extended indefinitely by adding further rings, while still preserving its Borromean quality (i.e. if any of the rings is cut, the whole chain falls apart). Lacan first takes up the Borromean knot in the seminar of 1972-3, but his most detailed discussion of the knot comes in the seminar of 1974-5. It is in this seminar that Lacan uses the Borromean knot as, among other things, a way of illustrating the interdependence of the three orders of the real, the symbolic and the imaginary, as a way of exploring what it is that these three orders have in common. Each ring represents one of the three orders, and thus certain elements can be located at intersections of these rings. In the seminar of 1975-6, Lacan goes on to describe psychosis as the unravelling of the Borromean knot, and proposes that in some cases this is prevented by the addition of a fourth ring, the SINTHOME, which holds the other three together. Borromean Knot, at the place where the three orders (Real, Symbolic and Imaginary) all intersect.

## Definition

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 ®, the symbolic (S), and the imaginary (I)—held together?

Indeed, the symbolic (the signifier) and the imaginary (meaning) seem to have hardly anything in common—a fact demonstrated by the abundance and heterogeneity of languages. Moreover, the real, by definition, escapes the symbolic and the imaginary, since its resistance to them is precisely what makes it real.

This is why Lacan identified the real with the impossible.) In psychoanalysis, the real resists, and thus is distinct from, the imaginary defenses that the ego uses specifically to misrecognize the impossible and its consequences.

If each of the three registers R, S, and I that make up the Borromean knot is recognized to be toric in structure and the knot is constructed in three-dimensional space, it constitutes the perfect answer to the problem above, because it realizes a three-way joining of all three toruses, while none of them is actually linked to any other: If any one of them is cut, the other two are set free. Reciprocally, any knot that meets these conditions is called Borromean. Note that the subject is now defined by such a knot and not merely, as with the cross-cap, as the effect of a cut (figure 8).

Unfortunately, this ideal solution, which could be considered normal (without symptoms), seems to lead to paranoia. Lacan considered this to be the result of failure to distinguish among the three registers, as if they were continuous, which indeed occurs in clinical work. Being identical, R, S, and I are only differentiated by means of a "complication," a fourth ring that Lacan called the "sinthome." By making a ring with the three others, the sinthome (symptom) differentiates the three others by assuring their knotting (figure 9).

In this arrangement, the sinthome has the function of determining one of the rings. If it is attached to the symbolic, it plays the role of the paternal metaphor and its corollary, a neurotic symptom.

Lacan also drew upon non-Borromean knots, generated by "slips," or mistakes, in tying the knots. These allowed him to represent the status of subjects who are unattached to the imaginary or the real and who compensate for this with supplements (Lacan, 2001). In such cases the sinthome is maintained.

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.

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## References

##### Index

- Lacan, Jacques.
*The Seminar of Jacques Lacan, Book XX: Encore, On Feminine Sexuality, The Limits of Love and Knowledge 1972-1973*. Trans. Bruce Fink. New York: W. W. Norton & Company, 1998. pp. 111, 123-36