Modular flow on the space of lattices

Stretch a lattice sideways and squash it vertically, forever. Its shape traces a geodesic on the modular surface, and the space it lives in turns out to be the complement of a trefoil knot.

The lattice

Take a lattice L in ℝ² with covolume 1 and apply g_t = diag(e^t, e^−t). That map preserves area, so L stays unimodular, but it gets smeared: long in x, thin in y. Any fixed basis you started with turns into two nearly parallel, enormous vectors. The lattice itself isn't degenerating. It's the same kind of object; you're just describing it with a bad basis.

Basis resets

So at each step the animation runs Gauss (Lagrange) reduction, the 2D version of the Euclidean algorithm: swap so v₁ is shorter, subtract the nearest integer multiple of v₁ from v₂, and repeat. That's a change of basis by a matrix in SL(2,ℤ), so the lattice is unchanged. When the cell flashes coral, it snaps to a fatter parallelogram. Those jumps are the whole reason the space of lattices is "folded."

The shape

Forget size and rotation and keep only the shape. Treat the basis vectors as complex numbers and let τ = v₂ / v₁. Changing basis by SL(2,ℤ) acts on τ by Möbius maps τ ↦ (aτ + b)/(cτ + d), and reduction lands τ in the standard fundamental domain |Re τ| ≤ ½, |τ| ≥ 1. That region, with its edges glued, is the modular surface ℍ/SL(2,ℤ). A handy fact: Im τ = 1/|v₁|², so the height tells you how short the shortest vector is.

Why the path is a geodesic

If the basis matrix has columns (a, c) and (b, d), then

Substituting w = i·e^2t gives τ = (b·w − d)/(a·w − c), a real Möbius map with determinant 1 applied to the imaginary axis. The imaginary axis is a geodesic in the hyperbolic plane, so its image is too. The flow on lattices is the geodesic flow on the modular surface. When the trail exits through one wall and reappears on the opposite one, that's the surface's gluing (τ ↦ τ + 1 or τ ↦ −1/τ), not a jump in the geometry.

What the buttons show

• Random lattice. A typical orbit wanders all over the domain, occasionally making tall trips up the cusp when some vector happens to be nearly vertical. This is ergodicity in action.
• Periodic orbit. The lattice built from the eigenvectors of [[2,1],[1,1]]. Since that matrix preserves ℤ², the flow returns to the same lattice every t = ln(φ²) ≈ 0.962, and the trail retraces one closed geodesic. Closed geodesics correspond to hyperbolic conjugacy classes in SL(2,ℤ).
• Square lattice. ℤ² contains the vector (0, 1), which shrinks by e^−t forever, so τ shoots straight up the cusp and never comes back. Lattices with a rational direction escape.

The trefoil

If you keep the rotation instead of throwing it away, the full space of unimodular lattices, SL(2,ℝ)/SL(2,ℤ), is homeomorphic to the 3-sphere minus a trefoil knot. The map sends a lattice to its Weierstrass invariants (g₂, g₃), and the trefoil is where the discriminant g₂³ − 27g₃² vanishes: the "lattices" that have collapsed. The cusp the square lattice falls into is the trefoil, seen from inside. Each periodic orbit becomes a closed loop in that knot complement, and Étienne Ghys proved that these modular knots are exactly the Lorenz knots, the same knots traced by periodic orbits of the Lorenz attractor. The 3D animations by Jos Leys that accompany Ghys's work show this same flow, drawn in the trefoil complement rather than on the flattened modular surface.

References

1. Étienne Ghys and Jos Leys, “Lorenz and modular flows: a visual introduction”, AMS Feature Column, November 2006. The best starting point. Illustrated tour of the lattice space, the trefoil, and the Lorenz connection; the source of the famous animations.
2. Étienne Ghys, “Knots and dynamics”, Proceedings of the International Congress of Mathematicians, Madrid, 2006, vol. I, pp. 247–277. The research-level account, including the proof that modular knots are Lorenz knots.
3. John Milnor, Introduction to Algebraic K-Theory, Annals of Mathematics Studies 72, Princeton University Press, 1971, §10. Where the space of unimodular lattices is identified with the complement of a trefoil knot.
4. Joan S. Birman and R. F. Williams, “Knotted periodic orbits in dynamical systems I: Lorenz’s equations”, Topology 22 (1983), 47–82. Introduces Lorenz knots via the template (knot holder) for the Lorenz flow.
5. Caroline Series, “The modular surface and continued fractions”, Journal of the London Mathematical Society (2) 31 (1985), 69–80. How geodesics on the modular surface encode continued fractions; a nice companion to the basis-reset picture.
6. Manfred Einsiedler and Thomas Ward, Ergodic Theory with a View towards Number Theory, Graduate Texts in Mathematics 259, Springer, 2011, ch. 9. A textbook treatment of the geodesic flow on SL(2,ℝ)/SL(2,ℤ) and its ergodicity.
7. Jean-Pierre Serre, A Course in Arithmetic, Graduate Texts in Mathematics 7, Springer, 1973, ch. VII. A short, classic derivation of the fundamental domain of the modular group.

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