|French: noeud borroméen
Lacan used the concept or image of the knot quite frequently. References to knots can be found in Lacan's work as early as the 1950s, but it is not until the ealy 1970s that Lacan begins to examine knots from the point of view of their topological properties. In the mid-1970s he tried to theorize the interrelation of the Symbolic, the Imaginary and the Real in terms of the topology of knots.
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 a 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 imaginary and the real; 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 to the right -- 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.
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 his view these orders are tied together in the form of a "Borromean knot". The "Borromean knot" is a linkage of three "string rings" in such a way that no two rings intersect. The structure of the knot is such that the cutting of any one ring will liberate all of the others. Lacan used the theory of knots to stress the relations which bind or link the Imaginary, Symbolic and Real, and the subject to each, in a way which avoids any notion of hierarchy, or any priority of any one of the three terms.)
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.