PILLOWCASE EXPLORER
● Interactive geometry
A VISUAL LABORATORY IN LOW-DIMENSIONAL TOPOLOGY

From a knot to a complex.

Cut. Restrict. Intersect. Follow the knot-version Atiyah–Floer picture, one geometric move at a time.

WORKED EXAMPLETrefoil 3₁Two-bridge model K(3, 1)
MOVE 01 / 05

A sphere makes the cut.

TANGLE DECOMPOSITION
Trefoil decompositionInteractive trefoil diagram.
WHAT YOU ARE SEEING

THE GEOMETRIC DICTIONARY
Step 1 of 5
≃
THE STRUCTURAL IDEA

Three-dimensional gauge theory leaves a two-dimensional shadow.
The shared boundary turns gluing into an intersection problem.

FROM PICTURE TO PAPER

Keep the geometry.
Follow the argument.

Instanton homology of knots

Full series & sources ↗
01 · CHARACTERS AND GRADINGS Traceless characters and instanton gradings for 2-bridge and (3,n) knots Bernd Johannes Wuebben · PDF ↗ 02 · PRETZEL KNOTS Instanton and pillowcase homology of the (−2,3,q) pretzel knots Bernd Johannes Wuebben · PDF ↗ 03 · BOUNDING COCHAINS Maurer–Cartan elements in the pillowcase and the Bar-Natan invariant for the (−2,3,q) pretzel knots Bernd Johannes Wuebben · PDF ↗ 04 · THE TORUS-KNOT DIFFERENTIAL The S-complex differential of torus knots Bernd Johannes Wuebben · PDF ↗ 05 · PILLOWCASE FLOER HOMOLOGY Pillowcase Floer homology of the torus knots T(3,n) Bernd Johannes Wuebben · PDF ↗
DIAGRAM DETAIL · SCROLL HORIZONTALLY ON SMALL SCREENS

A closer look.
QUICK REFERENCE

A small geometric vocabulary.
MODEL & REFERENCES

What is exact. What is schematic.

The curves and their intersections use explicit formulas. The knot drawing is a planar braid-closure model; the bigon animation is a separate local schematic. This is a teaching explorer, not an arbitrary-knot character-variety solver.

Coordinate convention (HHK)

a = i, b = eγki, c = eθki, d = e(θ−γ)ki; ba = cd.

Trefoil boundary curve

L₁(t) = (t, −2t), 0 ≤ t ≤ π.

Earring & holonomy-perturbed trivial tangle

L₀(β) = (β + π/2 + ε sin β, β + π/2 − ε sin β).

The slider uses 0.02 ≤ ε ≤ 0.22 radians, plus a degenerate ε = 0 comparison. Intersections solve 3β + 3π/2 + ε sin β = 2πn, n = 1, 2, 3. Labels x₁ and x₂ distinguish branches; they do not assert an absolute grading or the authors’ ± convention.
  1. Hedden–Herald–Kirk · Pillowcase and perturbations (2014) ↗

    Proposition 3.1, Theorem 7.1 and the two-bridge tangle calculation: coordinates, quotient and explicit curves.

  2. Hedden–Herald–Kirk · A Lagrangian–Floer theory (2018) ↗

    Surface Floer theory, relative gradings, and §7: the two-bridge complexes have zero differential and agree with reduced singular instanton homology.

  3. Cazassus–Herald–Kirk–Kotelskiy · The earring tangle (2022) ↗

    Why marking, immersed correspondences and bubbling matter in a broader gluing theory.

  4. Herald–Kirk · An endomorphism on immersed curves (2024) ↗

    Doubling and figure-eight operations associated with marked tangle correspondences.