Lacan Seminar 1974 topology borromean

The Seminar of Jacques Lacan, Book XXII: R.S.I.

Jacques Lacan

by Jacques Lacan

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Synopsis

Seminar XXII (R.S.I., 1974–75) represents Lacan's most sustained effort to ground psychoanalytic theory in the topology of the Borromean knot, advancing the argument that the Real, Symbolic, and Imaginary are not merely three conceptual registers but three material consistencies that only become genuinely three — achieve a "common measure" — through their Borromean knotting. Against Freud's geometrical (sack-based) topology, Lacan proposes that the knot is not a model or metaphor but a writing that directly supports the Real, and that each ring's three topological properties — consistency (Imaginary), the hole (Real), and ek-sistence (Symbolic) — are the proper structural correlates of the three registers. The seminar methodically redistributes Freud's triad of inhibition, symptom, and anxiety across RSI; reframes the Name-of-the-Father (and its plural, the Names-of-the-Father) as identical to the RSI triad itself and as a fourth element required to knot an otherwise loose three; reconceives the symptom as the particular way a parlêtre enjoys its unconscious; and redescribes feminine ek-sistence through the Not-all and the formula "The woman does not exist" as ones who are numerable but not universal. Throughout, Lacan treats the difficulty of mentally grasping the Borromean knot as itself the trace of primordial repression (Urverdrängung), and positions the analytic discourse — whose social bond he figures as the cartel of three-plus-one — as the first social formation to take the knot as its operative principle. The seminar closes by announcing the next year's work (Seminar XXIII, titled "4, 5, 6") on nomination as a fourth element and on the sinthome, signalling that the Borromean topology is not yet complete without a theory of what knots the knot itself.

Distinctive contribution

Seminar XXII's distinctive contribution to the Lacanian corpus lies above all in its elevation of the Borromean knot from illustrative figure to ontological operator. In earlier seminars (notably Seminar XI, Seminar XX, and the Écrits), the three registers are presupposed as a triad but their unity is never formally accounted for. Here Lacan asks the radical question: what makes them genuinely three? His answer — that only the Borromean structure provides a "common measure" without reducing the registers to a single substance — transforms RSI from a descriptive taxonomy into a topological claim about the nature of being for speaking beings (parlêtres). No other seminar devotes this density of attention to the material, manipulable reality of the knot as writing-of-the-Real, and no other text in the primary corpus develops the argument that the Borromean knot belongs to the Imaginary register (because grounded in three-dimensional space) while simultaneously serving as the form of the Real itself.

A second distinctive contribution is the seminar's radical pluralisation and structural redefinition of the Name-of-the-Father. Lacan declares that the Names-of-the-Father simply are the RSI triad — not a supplement from outside but the very act of naming that constitutes the three registers — while simultaneously arguing that the paternal function is not historically necessary but merely the tribal form that one particular civilisation (Freud's) gave to what is structurally just the fourth Borromean ring. This move opens directly onto the sinthome (Seminar XXIII), making Seminar XXII the essential hinge between the classic paternal-function seminars (Seminars III, V, XVII) and the late Joycean topology. Equally original is the seminar's sustained engagement with the topology of orientation: Lacan and his collaborators Soury and Thomé demonstrate that specifying (colouring) one ring of the Borromean knot necessarily generates two distinct orientated knots, a result with direct implications for sexuation and the irreducibility of dextro- vs. laevo-gyratory structure — a topological grounding for sexual difference that goes beyond anything in Seminar XX.

Main themes

The Borromean knot as writing of the Real rather than model or metaphor
The three registers (RSI) as material consistencies requiring topology, not geometry, for their common measure
The Names-of-the-Father as identical with the RSI triad and as a fourth ring that knots an otherwise loose three
Ek-sistence, consistency, and the hole as the three topological properties corresponding to Symbolic, Imaginary, and Real respectively
The symptom redefined as the parlêtre's particular mode of enjoying the unconscious — an untamed writing
Feminine ek-sistence and the Not-all: women as numerable ones, never as The
Primordial repression (Urverdrängt) as the hole that can never be grasped — figured by the mental resistance to imagining the knot
Phallic jouissance as the nodal term produced by the squeezing of the Symbolic ring, afflicting the male speaking being
The cartel's three-plus-one structure as socially grounded in the Borromean topology of the Name-of-the-Father
The transition toward nomination and the sinthome as the programme of Seminar XXIII (4, 5, 6)

Chapter outline

Introduction / Preliminary (19 November 1974) — p.1-6
Seminar 1: Tuesday 10 December 1974 — p.7-21
Seminar 2: Tuesday 17 December 1974 — p.22-33
Seminar 3: Tuesday 14 January 1975 — p.34-51
Seminar 4: Tuesday 21 January 1975 — p.52-66
Seminar 5: Tuesday 11 February 1975 — p.67-85
Seminar 6: Tuesday 18 February 1975 — p.86-100
Seminar 7: Tuesday 11 March 1975 — p.101-119
Seminar 8: Tuesday 18 March 1975 — p.120-133
Seminar 9: Tuesday 8 April 1975 — p.134-152
Seminar 10: Tuesday 15 April 1975 — p.153-170
Seminar 11: Tuesday 13 May 1975 — p.171-184

Chapter summaries

Introduction / Preliminary (19 November 1974) (p.1-6)

The seminar opens with Lacan's characteristic oblique framing: a strike at the university becomes the occasion for his first thesis, that the symptom belongs to the Real. Refusing to treat the strike as either celebration or political fact, Lacan insists it is an organised symptom — and the qualification 'organised' is immediately suspicious, since what is bad from the analyst's standpoint is precisely structure that has been conscripted by a discourse other than analytic discourse. He uses this occasion to announce the year's thematic: R.S.I., which can be read as an acronym for Real, Symbolic, Imaginary, or simply as letters whose equivalence is yet to be justified.

Lacan also raises, in this preliminary session, the question of the School and the pass (passe): what would it mean for analysts to constitute a set, to make themselves identifiable as such? This anticipates the later discussion of the cartel and of identification. The session frames the year's inquiry around the ethics of each discourse — those who are 'non-dupes' of its structure err precisely by believing they can stand outside it — and introduces the crowd itself as evidence that 'analytic discourse stirs you,' an ambiguous formulation that implicates the transference in the very gathering of the audience.

Key concepts: Symptom, Real, Analytic discourse, Non-dupes, Pass (passe), Discourse

Seminar 1: Tuesday 10 December 1974 (p.7-21)

Lacan opens by posing the problem of the 'common measure' of Real, Symbolic, and Imaginary: granted that they are different, what makes them genuinely three rather than merely disparate? The Freudian answer — a geometry of the sack (the second topography, with its Ego-lunula and Id-sack) — is dismissed as an inadequate imaginary topology that forces the inclusion/lesser-than confusion and cannot account for the capital-O Other as genuinely other rather than just external. The Borromean knot is proposed as the solution: it is the minimal topological structure that begins at three (breaking any one ring frees all the others), and it can be extended to any number while preserving this property, giving the three registers a common measure without reducing their heterogeneity.

Lacan then makes a claim he will defend throughout the year: the Borromean knot belongs to the register of the Imaginary, because it is supported by the number three and grounded in the three dimensions of space. This is counterintuitive — one might expect the knot to be assigned to the Real — but the argument is that the Imaginary is precisely what supplies three-dimensionality, and that it is by belonging to the Imaginary that the knot can support the triad RSI. The Real is introduced as 'strictly unthinkable,' the Symbolic as grounded in equivocation and writing, and the Imaginary as rooted in the body. The session also begins the redistribution of Freud's triad (Inhibition/Symptom/Anxiety) across the three registers, a redistribution that will be completed in subsequent sessions.

Key concepts: Borromean Knot, Real, Symbolic, Imaginary, Common measure, Topology, Inhibition, Symptom, Anxiety

Seminar 2: Tuesday 17 December 1974 (p.22-33)

This session consolidates the claim that the Borromean knot is not a model (which would be Imaginary in the pejorative sense of an illustration) but a writing that 'supports a real.' The distinction is crucial: a model mediates and represents; a writing directly instantiates. Each ring of the knot can take the form of an infinite straight line or a torus (ring of string), and the knot holds together not by resemblance among its elements but by the topological fact that each element is only knotted by virtue of a third. This nodality produces what Lacan calls 'phallic jouissance': the squeezing generated by the Symbolic ring's knotting with the others. Phallic enjoyment is thus not a biological given but a topological effect, produced at the nodal point where the Symbolic intersects with the other two registers.

Lacan introduces 'ek-sistence' as the key term for what is produced through knotting: to ek-sist is not merely to exist but to stand-out-from, to be defined by relation to a hole. Each of the three registers — Real, Symbolic, Imaginary — has a hole, and it is from these holes that ek-sistence is supported. Analytic experience, Lacan argues, is precisely the discovery that jouissance ek-sists — that it makes the Real by standing outside of consistency. The session thus distinguishes three aspects of the knot: consistency (what holds together), the hole (what ek-sistence circles around), and ek-sistence itself (the Real correlate of the knotted registers).

Key concepts: Borromean Knot, Ek-sistence, Phallic Jouissance, Real, Symbolic, Hole, Consistency, Torus

Seminar 3: Tuesday 14 January 1975 (p.34-51)

This session develops two related threads: the formal properties of extended Borromean chains and the question of what the knot implies for the concept of primordial repression. On the formal side, Lacan demonstrates that a Borromean knot of four (where any one ring, if cut, frees all others) requires a specific mode of knotting, and that the Peano successor function (n+1) provides an analogy for how the +1 ring holds the chain together — an analogy that will become literal when Lacan later identifies the fourth ring with the Name-of-the-Father. The unconscious itself is said to operate as an 'accountant' (comptable) that counts knots rather than items, which explains the structure of guilt (an account that can never be settled).

The second thread concerns the Urverdrängt, or primal repression. Lacan argues that the difficulty of mentally grasping the Borromean knot — the resistance of the 'mens' to imagining it — is not incidental but structural: it is the very trace of primordial repression, the hole that can never be reached. Nature, he says memorably, 'abhors the knot' — more precisely the Borromean knot, which is why geometry went through pyramids and cubes while ignoring the cord. The anxiety, symptom, and inhibition triad is revisited: anxiety is the bodily ek-sistence of jouissance (as demonstrated through the case of Little Hans, whose phobia gives body to the phallic burden); inhibition is arrest in the Symbolic; the symptom is the effect of the Symbolic in the Real. The phallus is not a natural genital drive but a symbolic imposition — the male parlêtre is 'afflicted' (aphligé) by it and can do nothing about it.

Key concepts: Borromean Knot, Repression, Urverdrängt, Anxiety, Symptom, Inhibition, Phallus, Jouissance, Name of the Father

Seminar 4: Tuesday 21 January 1975 (p.52-66)

Lacan takes up the formal topology of the knot with greater precision, introducing the distinction between the 'trivial' ring (which can be continuously deformed back into a simple circle) and the true knot (which cannot be so transformed). The Borromean knot is a 'second degree knot,' a knot of rings rather than a knot of cord, and its property of holding together without chaining is precisely what makes it the proper figure for the non-relationship of the sexual. The session also introduces objet petit a in this topological frame: as the cause of desire rather than its object, the small a marks an irrational, non-conjunctive gap between the One of the signifier and the One of meaning — a gap that cannot be written as a relationship.

The session pivots to the question of the symptom as writing. Lacan argues that the symptom is 'what does not cease to be written' from the unconscious — an 'untamed' writing that operates without convention. This is distinguished from the social symptom (as Marx used the term), though the two are not incompatible: the symptom in the private domain is the particular, irreducible way each subject's unconscious inscribes itself. The function of the symptom is written f(x), where x is 'what can be expressed of the unconscious by a letter,' and the letter is what isolates the identity-of-self-to-self from all quality. Language itself is described as 'only an ornure' — an ornamentation — while the discourse that really matters is the one that associates the subject determined by Being, i.e., by desire.

Key concepts: Symptom, Objet petit a, Desire, Writing, Signifier, Language, Sexual non-relationship, Topology

Seminar 5: Tuesday 11 February 1975 (p.67-85)

After reporting on his travels (Nice, Strasbourg, London), Lacan returns to the question of what it means for analytic interpretation to 'make a knot.' The interpretation does not merely deploy words; it is a 'saying' (dire) that has a different order of effect from word-use, and this effect is topological rather than semantic. Lacan introduces the distinction between the three consistencies (Imaginary, Symbolic, Real) as strictly equivalent — not hierarchically ordered — and argues that the standard philosophical move of privileging one consistency over the others is precisely what the Borromean topology corrects. Arab art's ornamental friezes and plaits, he notes, never produce a Borromean knot — evidence that certain 'consistencies' (in his technical sense) had not yet emerged as such.

The session introduces the topology of the torus as the figure for a consistency that has a hole without which there is no knot. For a construction to have non-imaginary consistency, there must be a hole — and this is the topology of the torus, which Lacan distinguishes from the sphere by the mode of writing that defines its auto-, homeo-, and automorphisms. The analytic interpretation is then framed as what produces a 'meaning effect' at the level of the knot, not at the level of the word: it tilts the bearing of meaning by operating on the topological structure of the three registers. A four-ring Borromean knot is introduced — requiring three independent toruses plus a fourth — which anticipates the subsequent elaboration of the Name-of-the-Father as fourth term.

Key concepts: Topology, Torus, Borromean Knot, Interpretation, Consistency, Hole, Name of the Father, Imaginary, Symbolic, Real

Seminar 6: Tuesday 18 February 1975 (p.86-100)

This session is more mathematically technical, devoted to the problem of how the Borromean knot resists geometric intuition — how 'flattening out' the knot is both necessary for demonstration and yet distorting. Lacan traces the origins of mathematical intuition to the cord (material consistency) rather than the line, point, or surface of classical geometry, arguing that geometry's more geometrico (Spinoza's model) is grounded in a mode of intuition that is fundamentally Imaginary and that topology corrects. The three dimensions of geometric space — point, line, surface — are shown to be fictions sustained by the Imaginary; the cord, by contrast, is real.

Lacan introduces the concept of ek-sistence as the Real dimension of the knot's structure: ek-sistence belongs to the field 'supposed by the rupture itself,' and this is why he can say that the mental knot ek-sists even if the mind cannot represent it. The session also takes up the obsessional's relation to death: for the obsessional, death is a parapraxis (failed act), because death is only approachable by an act, and to know that death is an act would require committing suicide knowingly — something very rare. This clinical aside is followed by the announcement of the next session's topic: the ek-sistence of women, who exist not as The but as numerable ones — a formulation of the Not-all that will be central to the later sessions on feminine sexuality.

Key concepts: Ek-sistence, Topology, Borromean Knot, Imaginary, Not-all, Obsessional, Anxiety, Feminine Sexuality

Seminar 7: Tuesday 11 March 1975 (p.101-119)

This session delivers one of the seminar's most decisive theoretical moves: the identification of the Names-of-the-Father with the RSI triad itself. The plural form — Names-of-the-Father, not the Name-of-the-Father — is not merely grammatical but structural: the Names-of-the-Father are 'the Symbolic, the Imaginary and the Real' in so far as they name, in the sense that the Bible's account of God teaching the speaking being to give a name to each thing is the mythological correlate of what Lacan is formulating topologically. Naming is thus not a predicate added to the three registers from outside but the very act by which each register acquires its 'dit-mansion' (the dwelling of the said).

Lacan then defines the Real as 'what ek-sists with regard to meaning' — as the expelled, the impossible, the anti-meaning, the aversion of meaning into anti-meaning. The Real is not inside or outside meaning but radically exterior to it, which is why it cannot be grasped by any semantic operation. Feminine ek-sistence is elaborated here through the formula that the Symbolic 'circles around an inviolable hole': what this means is that the hole of the Symbolic is what makes the Borromean knot Borromean, and feminine ek-sistence (as symptom for man) arises where the not-all resists phallic universality. The tribal God (the Name-of-the-Father in its religious form) is described as 'the quite useless complement' of the fourth ring's conjugation to the Symbolic — useful for man as a way of exiting the phallic predicament without loss, but structurally redundant once the knot is understood. Phallic jouissance is reframed as the signifier-index-1 that grounds all discourse, especially the discourse of the Master.

Key concepts: Name of the Father, Real, Symbolic, Imaginary, Ek-sistence, Feminine Sexuality, Not-all, Phallic Jouissance, Lalangue, Signifier

Seminar 8: Tuesday 18 March 1975 (p.120-133)

This session opens with a digression on the work of Lacan's collaborators Soury and Thomé, who discovered an error in a diagram from Seminar XX (Encore) and used it as the occasion to develop a new way of depicting Borromean knots that 'only come undone from one end.' Lacan treats this as evidence that his teaching has real effects — effects on mathematical work, not merely on clinical practice — and uses the occasion to reflect on the problem of objectivity. The Borromean knot, he argues, displaces the insoluble question of objectivity (the noumenon/phenomenon distinction) by proposing that 'the Real is that which determines ek-sistence' in the sense that the knot's topology forces a certain mode of turning-around — a cycle — that is more tractable than the noumenon while remaining irreducible to the phenomenon.

The session also advances the claim that the three consistencies (Imaginary, Symbolic, Real) are 'homogenised' by the knot — not made identical but placed in the same structural relationship to each other, each relating to the other two in the same way. This homogenisation is what makes the knot a proper figure for the non-existence of the sexual relationship: there is no way to write the sexual relationship as a logical-mathematical formula, because the three consistencies cannot be reduced to a single substance. Lacan also revisits the symptom as that 'which affects me as a symptom' — the RSI itself — and argues that what he is trying to do is give the 'I believe in it' of the symptom a different form of credibility, one grounded in topology rather than in meaning.

Key concepts: Borromean Knot, Topology, Consistency, Objectivity, Sexual non-relationship, Symptom, Real, Imaginary, Symbolic

Seminar 9: Tuesday 8 April 1975 (p.134-152)

Lacan opens with the observation that 'cogitation remains limed in the Imaginary of the body' — specifically in the figure of the sphere-and-cross (the body's self-image as circle-plus-internal-cross). This 'stickiness' of the Imaginary is what the Borromean knot is designed to escape: the knot is not naturally imagined as a knot, it presents itself to the imagination as 'two things that hook onto one another,' and one must actively work against this tendency to see it correctly. Lacan uses Joyce as an example of a writer maximally captured by the sphere-and-cross (St Thomas's influence via the Jesuits), and uses this to motivate the claim that the Borromean topology represents a new mode of thinking unavailable to traditional philosophical cogitation.

The session's mathematical core concerns the distinction between the 'cycle' (a track that encounters only consistency) and the 'hole' (what the cycle is centred on). The difference between two apparently equivalent Borromean configurations is located in this interplay: ek-sistence is the track, the going-around, while the hole is what is never reached. Lacan explicitly links this to the topology of death (unrepresentable, 'a stopper in the hole') and to the topology of topology itself (its resistance to geometric intuition, its emphasis on discontinuity, its notion of neighbourhood and accumulation point). The session closes with a detailed analysis of oriented Borromean knots: as soon as one ring is specified (coloured or made non-orientable by being rendered as an infinite straight line), two distinct orientated Borromean knots necessarily emerge — dextro- and laevo-gyratory — and this result cannot be transmitted in a message, implicating the irreducibility of sexual difference in topological terms.

Key concepts: Borromean Knot, Imaginary, Topology, Ek-sistence, Hole, Orientation, Feminine Sexuality, Real

Seminar 10: Tuesday 15 April 1975 (p.153-170)

This penultimate session is devoted to the topology of the non-relationship and to the cartel. Lacan returns to the two-torus model (from the earlier seminar on demand and desire) to show that the sexual non-relationship is not a failure of connection but the very structure knotted into being: two toruses can pass through each other's holes and make a chain, but this chain does not constitute a relationship in the logical-mathematical sense. The Borromean knot is not a chain; it is a knotting that holds together without chaining, and this is precisely what 'there is no sexual relationship' means at the topological level.

The discussion of the cartel then provides a social application of the same logic. The cartel's structure (three-plus-one) is not arbitrary but is grounded in the topology of the Name-of-the-Father as the fourth Borromean ring: with only three, one never knows which of the three is real (which one, if removed, would unknot the others); the fourth makes this visible. Nomination — the act of naming — is described as 'the only thing about which we can be sure that it makes a hole,' and it is for this reason that the cartel has four as its minimum. The Name-of-the-Father, in its function as nomination, is what holds the three registers together by introducing a hole that communicates consistency to all the others. Lacan also revisits his reading of Freud's three identifications (from Group Psychology) in the light of this topology, arguing that identification, love, and desire are each grounded in a different register of the knot's real Other.

Key concepts: Borromean Knot, Name of the Father, Cartel, Identification, Sexual non-relationship, Nomination, Torus, Symbolic, Real, Imaginary

Seminar 11: Tuesday 13 May 1975 (p.171-184)

The final session of the year announces the programme for Seminar XXIII (4, 5, 6) while consolidating the year's results. Lacan poses the question: why R.S.I. and not 1, 2, 3? His answer is that the number three does not simply designate the three registers but must be demonstrated as the Real — as the impossible — and this demonstration requires going beyond the three to the four, the five, and the six. The hole, which Lacan here names 'Soury's hole' (in tribute to his collaborator), is what searching amounts to: to search (chercher) is to circare, to circle around, and what is found is always a hole.

The session then takes up projective geometry (Desargues, Riemann) to argue that two straight lines at infinity are 'not knotted in a chain' — and that it is precisely by not being knotted two-by-two that they can be knotted by a third. This is the topological principle: the unknotted status of two terms is the condition of their being knotted by a third, and this generalises to the fourth term — nomination — distributed across the three registers. Each mode of nomination corresponds to one of the three registers: inhibition (Imaginary nomination), anxiety (Real nomination), and symptom (Symbolic nomination), and these converge on the Name-of-the-Father as the fourth that knots all three. The seminar ends by pointing toward the sinthome — the particular knotting that each speaking being makes of its own real — as the object of the coming year's work.

Key concepts: Borromean Knot, Nomination, Name of the Father, Sinthome, Topology, Real, Inhibition, Anxiety, Symptom, Projective geometry

Main interlocutors

Sigmund Freud, Beyond the Pleasure Principle
Sigmund Freud, Inhibitions, Symptoms and Anxiety
Sigmund Freud, The Interpretation of Dreams
Sigmund Freud, Schreber case
Freud, Group Psychology and the Analysis of the Ego
Jacques Lacan, Seminar XX (Encore)
Jacques Lacan, Seminar XXI (Les Non-Dupes Errent)
Jacques Lacan, Seminar XXIII (The Sinthome)
Jacques Lacan, Seminar XI
Jacques Lacan, Seminar X
Jacques Lacan, Seminar VII
Jacques Lacan, Seminar III
Jacques Lacan, Seminar I
Jacques Lacan, Écrits
Jacques Lacan, Rome Discourse
René Descartes, Rules for the Direction of the Mind (Regulae ad Directionem Ingenii)
Hegel, Phenomenology of Spirit
Gottlob Frege, Grundlagen der Arithmetik
Goethe, Faust
Plato, Cratylus
Aristophanes, Lysistrata
Velázquez, Las Meninas
Michel Thomé and Pierre Soury (collaborators on Borromean knot topology)
Giuseppe Peano (axiomatic arithmetic)

Position in the corpus

Seminar XXII occupies a pivotal hinge position in the Lacanian corpus: it is the culmination of the trajectory that runs from Seminar XI's theorisation of the four fundamental concepts through Seminar XX's formulas of sexuation and the Not-all, and it is the immediate predecessor of Seminar XXIII's theory of the sinthome as a fourth Borromean ring. Readers coming from Seminar XX (Encore) will find RSI to be its topological deepening: where Encore introduced lalangue, jouissance féminine, and the Not-all at the level of logic and linguistics, Seminar XXII grounds these same results in the material topology of the knot. Those coming from the paternal-function seminars (Seminars III, V, XVII) will find a dramatic revision: the Name-of-the-Father is no longer the singular exception that institutes the Symbolic but one possible fourth ring among many, its privilege merely tribal. Seminar XXII should be read after Seminar XX and before Seminar XXIII; familiarity with the basic RSI distinction (available from Seminar XI or the Écrits) is a prerequisite.\n\nWithin the broader Lacanian-Žižekian-Millerian interpretive corpus, Seminar XXII is often cited but rarely read closely, partly because it has not been officially published in French and circulates only in unauthorised transcriptions. Its topology is more demanding than anything in the earlier seminars, and its clinical payoff is less immediately visible than Seminar XI or Seminar XVII. It shares theoretical ground with Seminar XXIII (the sinthome, Joyce, the fourth ring) and with the late Lacan's work on nomination and the parlêtre, and it is indispensable background for any serious engagement with the concept of the sinthome. For readers interested in Lacan's philosophy of mathematics, the sessions on Peano, Frege, Desargues, and Riemann make this seminar uniquely valuable.