False Hole

The "false hole" (faux-trou) is a topological concept introduced in Seminar XXIII to name the specific kind of hole produced when two rings of a Borromean chain are conjoined. Unlike the hole that belongs properly to each individual ring, the false hole is neither the hole of the one nor the hole of the other: it is a third, emergent void that comes into existence only through their conjunction, located "in the middle" as an effect of the relation rather than of either term. The "false" qualifier does not mean illusory or non-existent; rather, it marks that this hole is not grounded in a pre-given topological substance of either circle but is, precisely, produced — a relational artifact. Lacan deploys this concept to explain why the Borromean knot is irreducible to dyadic combination: the conjunction generates something new that cannot be reduced back to its components.

Lacan then immediately promotes the Phallus as the "verifier" of this false hole, which is to say: the Phallus is the signifier that confirms and attests to the reality of this emergent void. In doing so, it constitutes the Real — not as something pre-given but as what is verified or guaranteed by the phallic function. This move connects the topological register (the knot, the false hole) to the sexuating register (the Phallus, the relation between the sexes), and extends into lalangue and the sinthome (Joyce's in particular): all are understood as structures in which a produced, "false" hole — something that emerges from conjunction and lacks a stable ground in either partner — must be verified, knotted, or repaired by a supplementary function.

The concept of the false hole lives in Seminar XXIII (jacques-lacan-seminar-23-cormac-gallagher), Lacan's seminar on James Joyce and the sinthome — the moment in his teaching where the Borromean Knot receives its most sustained elaboration. It is a specification of the Borromean Knot's topology: whereas the Borromean Knot's canonical definition foregrounds three rings whose irreducible interdependence holds the registers of RSI together, the false hole drills into what happens at the dyadic level within that triadic structure — what is produced in the zone of intersection between any two rings. In this sense, the false hole is a technical refinement that names the ontological status of what fills those overlapping regions. It answers a question the general Borromean account leaves open: what kind of entity is the overlap, and how is it related to the Real?

In relation to the other cross-referenced concepts, the false hole sits at the intersection of several axes. With respect to the Imaginary (which carries the topological property of consistency in the late seminars), the false hole is neither a purely imaginary nor a purely symbolic object: it is an emergent, relational void that requires the Phallus as verifier — connecting it to the Real. With respect to Lalangue, the concept extends the topological insight: lalangue is itself a site where two registers (the material phonemic body and the symbolic chain) conjoin and produce an excess that belongs to neither — a "false hole" at the level of language. With respect to Sexuation and the Not-all, the false hole figures the structural incompleteness of the sexual relation itself: the "relation between the sexes" that does not exist as a symmetrical whole is precisely that which the Phallus must come to verify without being able to ground in any natural or pre-given unity. The false hole is thus the topological name for what remains ungrounded but real in the conjunction of two terms.

Seminar XXIII · The SinthomeJacques Lacan · 1975 (p.142)

what is proper to it and what is what I will call the false hole (faux-trou)... the fact that one can with one circle by uniting another with it, make this hole which consists in what is happening there, in the middle and which is neither the hole of one nor the hole of the other, this is what I call the false hole.

The phrase "neither the hole of one nor the hole of the other" is theoretically loaded because it establishes that the false hole has no ontological ground in either constitutive ring — it is purely relational and emergent, produced only by conjunction ("by uniting another with it"). The localization "in the middle" further refuses any attribution of the hole to a pre-existing substance, making the false hole a paradigm case of the Lacanian Real as that which is produced rather than found — verified, as Lacan says, by the Phallus rather than given in advance.