Longitudinal Cut

The Longitudinal Cut is a topological operation performed on a reversed torus — one that has been turned inside-out — in which the cut follows the concentric (longitudinal) axis of the toric surface. The decisive theoretical property of this cut is that it dissolves the Borromean knot: by sectioning the reversed torus along this axis, the rings that compose the Borromean structure are freed from one another. This contrasts sharply with the transversal cut, which runs perpendicular to the longitudinal axis and fails to dissolve the knot. The distinction is not merely geometric but structural: the direction of the cut determines whether the interdependence constitutive of the Borromean linkage is broken or preserved.

The argument extends to a six-fold Borromean structure, where Lacan posits that the consequences of reversal — and therefore the efficacy of a given cut — differ depending on the arrangement of the rings. This implies that the longitudinal/transversal distinction is not absolute but is conditioned by the topology of the specific knot-configuration under analysis. The concept thus belongs to the late Lacanian project of using knot theory to formalise the conditions under which the RSI triad (or its more complex derivatives) can be dissolved or sustained — making the cut a concrete operator within topological writing rather than a metaphor.

Within jacques-lacan-seminar-25, the Longitudinal Cut is a specification of the broader topological apparatus that Lacan had been developing throughout his late seminars. It presupposes the Reversed Torus as its object of operation and is defined entirely by its contrast with the Transversal Cut. The concept is thus intelligible only within the framework of Topology as Lacan deploys it: not as an illustration but as a formal writing whose operations directly model structural relations. As the canonical definition of Topology notes, "A topology is always founded on a torus, even if this torus is at times a Klein bottle" — the torus is the foundational object, and cutting procedures on it are therefore operations on structure itself.

In relation to the Borromean Knot, the Longitudinal Cut functions as a dissolution operator: it undoes the triadic interdependence that the knot formalises. Where the Borromean Knot models the RSI registers as holding together only so long as all three rings are present, the Longitudinal Cut identifies the precise geometric condition under which that hold is released. This makes it a kind of inverse of the knotting function — not the fourth ring that repairs or sustains (as the sinthome does in Seminar XXIII), but the axial incision that undoes. The extension to a six-fold Borromean structure further suggests that Lacan is exploring how structural complexity (more rings, more arrangements) modulates the effect of topological operations, consistent with the late-Lacanian programme of writing the conditions of clinical stabilisation and dissolution in purely formal terms.

Seminar XXV · The Moment to ConcludeJacques Lacan · 1977 (p.12)

the cut (2) that I have just made dissolves the Borromean knot... I can call the other sense transversal (1). The transversal does not free the threefold torus but on the other hand the longitudinal frees it.

The quote is theoretically loaded because it identifies "dissolves" and "frees" as the operative verbs, making the Longitudinal Cut not merely a geometric description but a structural event: the Borromean interdependence — the very condition of RSI knotting — is undone by this axis of incision. The explicit opposition between "transversal" (which does not free) and "longitudinal" (which does) inscribes directionality as the decisive formal variable, grounding what might otherwise seem a purely mathematical distinction in the Lacanian stakes of what it means to break, or not break, a knot.