Ronds

a workbench for Borromean rings and knots

Presets:

Alexander Polynomial:

Gauss:
Dowker-Thislethwaite:

Rings:

Force Smoothness: |

Move whole rings: | Show fields (mise à plat): | Drag Neighboring Points: | Allow crossings (uncheck to enforce isotopy — permit only Reidemeister moves, no crossing changes while dragging):

Additional Invariants, calculated once:

Styling:
Gap Width:
Stroke Width:
Show Intersections: Drag Handles Independently:

Exporting:
Importing:

From Knottingham JSON:

Manual

Ronds is a workbench for manipulating 2-4 linked rings, in the spirit of Lacan's Borromean clinic (R, S, I, and the fourth ring Σ/sinthome). It is a fork of Knottingham, a tool for drawing and manipulating general knot diagrams. To start, you may want to follow these steps:

Load a starting diagram with one of the preset buttons above the canvas (Borromean rings, the "faulted" Borromean lapsus, a chain, a Hopf link, a trefoil, the Borromean with a droite infinie, or the 4- and 5-ring Borromean chains).
Rearrange rings by:

Dragging a ring anywhere on its stroke to move the whole ring, while "Move whole rings" is checked (this is the default)
Unchecking "Move whole rings" to go back to editing individual nodes: clicking segments to add nodes, dragging nodes to move them, Shift+Click to remove a node, Control+Click to smooth a node
Moving and rotating the selected ring with the WASD+EQ keys, mirroring everything with M

Click on a crossing to flip which ring goes over — crossings are always clickable, so on the Borromean you can turn it into the faulted/lapsus version, or any other arrangement, by hand
Use the Rings panel to rename a ring, recolor it implicitly through its palette slot, remove it (try removing one ring from the Borromean preset — the other two spring free, and Undo restores it with its name and color intact), or add a new ring by drawing one
Undo with the button or the Z key
Allow crossings is on by default, so you can freely change the knot. Uncheck it to enforce isotopy (revert non-Reidemeister moves): dragging then keeps the diagram equivalent to itself, so you can rearrange the rings without changing the link type. Clicking a crossing always flips it either way, since that is a deliberate change.

Cutting a ring — la coupure. The "✂ cut" button in the Rings panel opens a ring: a short arc is removed and the two free ends are capped with dots. An open strand can always slip free, so a cut ring is excluded from the linking numbers and the other invariants (though its crossings stay drawn until you drag it out or mend it). Cutting any ring of the Borromean preset frees the other two — that is the Borromean property. Contrast with the faulted Borromean knot (the lapsus): cut I there and R and S stay linked, lk(R,S) = ±1. "Mend" re-closes the ring as it was before the cut.

The droite infinie. In the "Borromean (droite infinie)" preset, the third ring is drawn as an infinite straight line — a ring closed at infinity, as Lacan draws it in RSI. On the canvas it is a very large loop whose return arc lies far off-screen; it behaves like any other ring.

Mise à plat. The flattening of the knot is not a degradation — Lacan calls the mise à plat the basis of all figuration of the knot. It is on the flattened diagram that he displaces Inhibition, Symptom and Anxiety "onto three planes," and inscribes the fields: a at the central lock-point, JA (jouissance de l'Autre) between Real and Imaginary, JΦ (jouissance phallique) between Real and Symbolic, sens between Symbolic and Imaginary, with ex-sistence hatched outside the consistencies along the infinite line, where Φ — the phallus that ex-sists — is written (RSI, 10 Dec 1974, 17 Dec 1974, 13 & 21 Jan 1975; cf. Le Sinthome, 16 Dec 1975: "the flattened-out overlap of the circle of the Symbolic with the circle of the Imaginary — which is meaning"). Turn on "Show fields" to see the planar regions of any diagram; the RSI (mise à plat) button loads the canonical figure with all inscriptions. Click inside a field for a one-line gloss.

Multi-component links (including all the ring presets) show a per-component Gauss code and the pairwise linking numbers, and the Jones and HOMFLY polynomials as well as orthogonalization work through the diagram's PD code. The Alexander polynomial and the Dowker-Thistlethwaite code are only computed for single knots (like the trefoil preset).

Credit

Ronds is built on top of Knottingham, created by Lennart Finke and Edmund Weitz and released under the MIT licence. All of the underlying knot-diagram machinery — crossing detection, invariant computation, Bezier editing, orthogonalization, SVG/TikZ export — is theirs. You can read about it in their preprint here or the finished paper in the IEEE Transactions on Visualization and Computer Graphics. If this tool helped your research or teaching, please cite the original work as:

          @article{finke2024,
            author={Finke, Lennart and Weitz, Edmund},
            journal={IEEE Transactions on Visualization and Computer Graphics},
            title={A Phenomenological Approach to Interactive Knot Diagrams},
            year={2024},
            volume={30},
            number={8},
            pages={5901-5907},
            doi={10.1109/TVCG.2024.3405369}}

Knottingham is open source - you can read and contribute to the original code over here.

The Jones and HOMFLY polynomial are calculated with SageCellMath. To help them cover server costs incurred through websites like this, they accept contributions and donations here.