Ran₁(S¹) = singletons, a trefoil
the loop γs at time s
Ran₂(S¹) = pairs, a Möbius band
disk swept so far
the loop γs at time s
Ran₂(S¹) = pairs, a Möbius band
disk swept so far
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the homotopy extruded{(θ, x, s) : x ∈ γs(θ)} ⊂ S¹θ × S¹x × [0,3]s, the torus drawn as a square with its opposite edges secretly glued: θ to the right, x up, s into the depth. The sheets are the strands, merging where points of γs(θ) coincide. The slice at depth s is the strand diagram on the right.
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the loop γs as strands: θ → γs(θ) ⊂ S¹
the configuration γs(θ) at the marker
What is drawn. A point of Ran₃(S¹) is a nonempty subset of the circle with at most three elements. Following Mostovoy, such a subset is the set of directions (doubled angles) from the vertices of an acute triangle to its circumcentre; the triangle generates a lattice in ℂ, and Weierstrass’ (g₂, g₃) of that lattice, normalised, is a point of S³ ⊂ ℂ². Singletons ↔ degenerate lattices ↔ the (2,3) torus knot g₂³ = 27g₃²; pairs ↔ rectangular lattices ↔ a Möbius band whose core is the circle of antipodal pairs (square lattices). The picture is a stereographic projection of that S³ from a point on the circle of equilateral triples (hexagonal lattices).
The homotopy γs, s ∈ [0,3], starts at the diagonal γ₀(θ) = {θ}. Stage 1 (s ≤ 1): {θ, θ + sπ} — a second point grows out of the first; the loop slides across the Möbius band and lands on its core, traversed twice. In Ran₂ this is as far as one can go: the boundary of a Möbius band is twice its core, so γ₀ is not null-homotopic there. Stage 2 (1 ≤ s ≤ 2): still inside the band, the doubled core is slid to the doubled loop m(t) = {p, p + t}, which keeps one point fixed at p = −π/2. Stage 3 (2 ≤ s ≤ 3): using a genuine third point, m contracts: {p, p + 2πr, p + 2πr + (1−r)t} with r = s − 2 — one point sits at p, a second runs once around the circle, and the third sweeps the arc between them. The image of this last stage, the set of subsets containing p, is a dunce hat {p} ∪ Ran₂(S¹) inside Ran₃(S¹).
Or straight to the origin. The other setting of contract replaces stages 2–3 by the obvious move in the picture: the doubled core is the unit circle of the horizontal plane, so shrink it radially across the flat disk it bounds. That disk is explicit in S³ (lattices with g₃ purely imaginary) but not in configurations: each of its points is sent back to a subset of S¹ by inverting the whole construction — recover j, solve j(τ) by Newton’s method, rebuild the acute triangle — which is how the strand diagram is filled in. Its centre is one equilateral triple, and on the way in the disk passes through three singletons of the trefoil.
A closed-form model of the flat disk. Numerically the flat disk has a simple combinatorial shape, and the third setting of contract draws an explicit formula with that shape. Let ck = π/6 + 2πk/3 be the equilateral triple at the centre. The disk meets the Möbius band in three arcs, the pairs {ck ± g/2}, g ∈ [0, π], running from the core (g = π) to the singletons ck (g = 0); these arcs cut each loop into three sectors, carried into one another by the ℤ/3 symmetry (rotate the circle by 2π/3, the disk by 4π/3). In the sector from the arc at cm+1 to the arc at cm+2, with u ∈ [0, 1], γ = { cm+1 − g/2 + u(g − 4π/3), cm+1 + g/2 + u(2π/3 − g) ± σ(g) sin πu }: a lone point runs backwards from cm+1 − g/2 to cm+2 + g/2 while a pair runs forwards from cm+1 + g/2 to cm+2 − g/2, opening and closing on the way. At a sector boundary the pair has closed and the lone point is about to split — the same set either way, which is the fold {a, a, b} = {a, b, b} of Ran. At g = 0 every sector runs from singleton to singleton, and from there the three strands (displacements 2π/3 − 4πu/3, 2πu/3 + σ sin πu, −2π/3 + 2πu/3 − σ sin πu from their homes cm, cm+1, cm+2) are scaled linearly to zero. With σ(g) = 0.136π·(1 − g/π) the model stays within about a tenth of the picture’s unit of the flat disk.
The homotopy γs, s ∈ [0,3], starts at the diagonal γ₀(θ) = {θ}. Stage 1 (s ≤ 1): {θ, θ + sπ} — a second point grows out of the first; the loop slides across the Möbius band and lands on its core, traversed twice. In Ran₂ this is as far as one can go: the boundary of a Möbius band is twice its core, so γ₀ is not null-homotopic there. Stage 2 (1 ≤ s ≤ 2): still inside the band, the doubled core is slid to the doubled loop m(t) = {p, p + t}, which keeps one point fixed at p = −π/2. Stage 3 (2 ≤ s ≤ 3): using a genuine third point, m contracts: {p, p + 2πr, p + 2πr + (1−r)t} with r = s − 2 — one point sits at p, a second runs once around the circle, and the third sweeps the arc between them. The image of this last stage, the set of subsets containing p, is a dunce hat {p} ∪ Ran₂(S¹) inside Ran₃(S¹).
Or straight to the origin. The other setting of contract replaces stages 2–3 by the obvious move in the picture: the doubled core is the unit circle of the horizontal plane, so shrink it radially across the flat disk it bounds. That disk is explicit in S³ (lattices with g₃ purely imaginary) but not in configurations: each of its points is sent back to a subset of S¹ by inverting the whole construction — recover j, solve j(τ) by Newton’s method, rebuild the acute triangle — which is how the strand diagram is filled in. Its centre is one equilateral triple, and on the way in the disk passes through three singletons of the trefoil.
A closed-form model of the flat disk. Numerically the flat disk has a simple combinatorial shape, and the third setting of contract draws an explicit formula with that shape. Let ck = π/6 + 2πk/3 be the equilateral triple at the centre. The disk meets the Möbius band in three arcs, the pairs {ck ± g/2}, g ∈ [0, π], running from the core (g = π) to the singletons ck (g = 0); these arcs cut each loop into three sectors, carried into one another by the ℤ/3 symmetry (rotate the circle by 2π/3, the disk by 4π/3). In the sector from the arc at cm+1 to the arc at cm+2, with u ∈ [0, 1], γ = { cm+1 − g/2 + u(g − 4π/3), cm+1 + g/2 + u(2π/3 − g) ± σ(g) sin πu }: a lone point runs backwards from cm+1 − g/2 to cm+2 + g/2 while a pair runs forwards from cm+1 + g/2 to cm+2 − g/2, opening and closing on the way. At a sector boundary the pair has closed and the lone point is about to split — the same set either way, which is the fold {a, a, b} = {a, b, b} of Ran. At g = 0 every sector runs from singleton to singleton, and from there the three strands (displacements 2π/3 − 4πu/3, 2πu/3 + σ sin πu, −2π/3 + 2πu/3 − σ sin πu from their homes cm, cm+1, cm+2) are scaled linearly to zero. With σ(g) = 0.136π·(1 − g/π) the model stays within about a tenth of the picture’s unit of the flat disk.