← mino.mobi

temperley–lieb

the algebra of non-crossing loops · tile it, then wrap it

↑ geometry pack · sibling of meander · runner · viazovska · torus craft: torus.mino.mobi

Fill a grid with the two Temperley–Lieb tiles — each cell connects its four edges with a non-crossing pair of arcs. The arcs link up across cells into loops, and every closed loop is worth a weight δ. Now choose how the grid's edges are glued: a flat disk, a cylinder (left meets right), or a torus (both pairs glued). On the cylinder and torus, loops can wind around and never close up inside the picture — those are the non-contractible loops of the affine Temperley–Lieb algebra. Click any tile to flip it.

boundary
columns · L
6
rows · M
6
loop weight · δ
1.41
Special δ = 2·cos(π/(k+2)) — 1, √2, φ, √3, … — are where TL has its famous quotients (the Jones subfactors, the ADE classification). δ = 2 is the boundary between "small" and generic. Switch to the torus tab to see this same configuration wrapped onto a donut.

The loop tiles are the Q-state Potts model in disguise — loop weight δ = √Q, so δ = √2 is exactly the Ising model. Here the δ slider is a thermostat: we randomly flip tiles and accept each flip with Boltzmann probability min(1, δΔloops), sampling the loop gas at that fugacity. The tribes are the loops — and which tribe a tile joins is decided by global connectivity, not its orientation. Hit run and watch the tribes reorganise. (Torus boundary, so every tribe closes; winding tribes are coloured hot.)

loop fugacity · δ (Q = δ²)
1.41
grid · L × M
10
speed · updates / sec (lower = calmer)
3
δ = 1 is free (infinite temperature — every tiling equally likely). δ > 1 rewards more tribes (many small loops); δ < 1 rewards fewer (a few big, often winding, loops). At δ = √2 the loops are the Fortuin–Kasteleyn cluster boundaries of the Ising model — the tribes you're watching are FK clusters.

Now a real temperature axis. This is the Ising model — Potts at Q = 2, the δ = √2 point of the thermal tab — driven by the Swendsen–Wang cluster thermostat: open bonds between aligned neighbours with probability 1 − e−2/T, then flip whole tribes (the Fortuin–Kasteleyn clusters) at once. Cool through the exact critical temperature Tc = 2 / ln(1 + √2) ≈ 2.269 and watch the tribes diverge: a froth of tiny tribes when hot, one fractal spanning tribe at Tc, a single frozen continent below. Hit anneal to ramp the temperature and ride through the transition. (Torus boundary.)

temperature · T (Tc ≈ 2.269)
2.269
view
grid · L × L
48
speed · updates / sec (lower = calmer)
3
The domain loops view draws the walls between up and down regions — the Peierls contours, the same loops as the rest of this page — in a fixed colour, with the majority phase gauge-pinned so it never swaps tint. Swendsen–Wang flips whole correlated tribes per step, so it stays fast even at Tc where single-spin flips freeze. The biggest-tribe fraction is the order parameter: it climbs from ~0 to ~1 across Tc.

The Potts model is the Ising model with Q colours instead of two: each site picks one of Q states, and neighbours that match lower the energy. Q = 2 is exactly Ising; Q = 3, 4, … are its colourful cousins — and the loop weight is δ = √Q, so this is the very same family as the thermal tab, now dialled to any integer Q. Same Swendsen–Wang thermostat: bond matching neighbours with probability 1 − e−1/T, then recolour whole domains at once. Each Q has its own exact critical temperature Tc = 1 / ln(1 + √Q). Cool through it and watch Q-coloured continents freeze out. For Q ≤ 4 the freeze is continuous; for Q ≥ 5 it turns abrupt (first-order) — crank Q up, sit near Tc, and you can catch ordered and disordered regions coexisting. (Torus boundary.)

colours · Q (Q = 2 is Ising)
3
temperature · T
0.995
view
grid · L × L
48
speed · updates / sec (lower = calmer)
3
Order parameter is the Potts magnetisation m = (Q·fmax − 1)/(Q − 1), where fmax is the fraction in the most common colour: m ≈ 0 disordered, → 1 frozen. The domain-loops view draws the walls between unlike colours in a fixed accent — a Q-colour Peierls picture, calm by default at ≤ 3 updates/sec.

The phase diagram ties the thermostats together. Sweep the temperature from cold to hot and, at each T, let Swendsen–Wang equilibrate and measure the order parameter — the Potts magnetisation m. The curve falls from 1 (frozen, ordered) to 0 (disordered) as you cross the exact critical temperature Tc = 1/ln(1+√Q) (dashed line). For Q ≤ 4 the fall is continuous; for Q ≥ 5 it drops off a cliff — a first-order transition. Pick Q, hit run, and watch the curve draw itself.

colours · Q (Q = 2 is Ising)
2
grid · L × L (smaller = faster)
24
Blue is the magnetisation m; faint orange is the energy per site. The sweep heats cold→hot, carrying the lattice from each T to the next (adiabatic), so the curve tracks the equilibrium branch. Finite L rounds the corner off — bump L up to sharpen the transition, down for speed.

The tiling projected onto a real torus (the gluing the geometry pack's toroidal games use) — and now you can run the loop-gas thermostat right here on the donut. Hit run and the same δ-weighted Metropolis dynamics from the thermal tab reshuffle the loops live in 3D. Contractible loops sit in a patch; non-contractible loops wind through the hole or around the tube and can't be shrunk away. Pull δ down and watch winding loops appear and thread the donut — that winding is the whole point of the affine algebra. Drag to rotate.

loop fugacity · δ (Q = δ²)
1.41
tube radius · r/R
0.42
speed · updates / sec (lower = calmer)
3
Run the thermostat here, or edit on the tiles tab (torus mode) and hit ↻ from tiles to mirror it. A loop that circles the hole is winding in one direction; one that circles the tube winds in the other. The pair of winding numbers is the loop's homology class on the torus.

A continuous path that visits every square of the torus is a Hamiltonian path. At each interior cell it either goes straight or turns — and tortuosity τ = turns / (interior cells) measures how wiggly it is, maxed out at τ = 1 when every cell is a turn. That all-turns limit is exactly the loop picture this whole page runs on: every arc is a 90° turn. Maximising turns is a combinatorial optimisation, so we throw the Monte-Carlo toolset at it: simulated annealing with backbite moves (reconnect one endpoint to a lattice neighbour, reversing the tail — the standard ergodic move on Hamiltonian paths) that minimises the count of straight cells. Hit run and watch the snake crumple into a maximally twisty path.

grid · L × L
14
anneal temperature · τT (auto-cools)
1.20
speed · updates / sec (lower = calmer)
4
Red dots mark the straight cells — the defects from a perfectly twisty path; squares are the two endpoints. Lower τT greedily hunts turns, higher τT shakes the path loose to escape local traps. "Best" keeps the twistiest path seen so far. Wraps across the torus seam show as stubs at opposite edges.

A parity law falls out: all-turns (τ = 1) is reachable only when L is even. For odd L the minimum number of straight cells is exactly L − 1 — an all-turns path forces moves to alternate horizontal/vertical, which tiles each column with length-2 vertical steps, impossible for odd L. The annealer hits this bound on the nose every run.

Temperley–Lieb is an algebra: its elements are diagrams on n strands, and you multiply them by stacking. The generators ei cap strands i and i+1. Build a word by clicking generators; the diagram reduces to a basis picture times δ(loops), and the defining relations fall out as pictures.

strands · n
4
append generator
check a relation
The "close (trace)" button glues each top point to the bottom point below it and counts the loops — the Markov trace. Pairing two such diagrams is exactly an entry of the meander matrix on /meander/.

Pictures you can multiply

The Temperley–Lieb algebra TLn(δ) is an algebra whose elements are diagrams. Fix n strands. A basis diagram connects n points on the bottom of a box to n points on the top with non-crossing strands. There are exactly Cn of them — the Catalan number, the same count that runs every row of arches on /meander/. A TL diagram and a meander arch system are the same combinatorial animal in different costumes.

Two rules make it an algebra:

The relations are pictures

Generated by 1 and e1, …, en−1 (each ei caps strands i, i+1 top and bottom), with three relations you can see in the algebra tab:

That's the whole presentation. dim TLn = Cn (1, 2, 5, 14, 42, 132, …).

Tiles, and the loop model

A word in the generators is a tiling of a rectangle by two tiles — a pass-through and a turn. The tiles tab lets you set each cell freely, so you're building a configuration of the completely-packed loop model on a lattice: the very model behind the Potts model, the six-vertex model, and the Jones polynomial. Counting loops weighted by δ is the algebra's trace. At δ = 1 each loop counts once, and the single-loop configurations are precisely the meanders.

Does it wrap? — the affine / annular algebra

Planar TL lives in a disk (a flat box). Roll the box into a tube so strand n+1 is strand 1 again and you get the affine (a.k.a. periodic or annular) Temperley–Lieb algebra. Strands now wind around the cylinder, and a new kind of loop appears: a non-contractible one that circles the tube and can't be shrunk away. These winding loops usually carry their own weight, separate from δ. This is the algebra of the periodic XXZ spin chain and the loop/Potts model on a cylinder — the setting where you read off conformal field theory data (central charge, the operator spectrum). It is the same loop-gas / Coulomb-gas family whose conformal weights, gravitationally dressed, produce the quantum-gravity exponent on the meander page.

Glue the other pair of sides too and you have a torus: loops can wind in two independent directions, the partition function becomes modular invariant, and the loop model's CFT identity is fully exposed. The torus tab shows your tiling on the donut so you can watch a loop thread the hole. So: yes it wraps; yes the strands become loops; and how you wrap — disk → cylinder → torus — is exactly the dial that turns a combinatorial toy into integrable statistical mechanics.

The tiles are Catalan-simple. The gluing is where the physics lives.

The Markov trace, and back to meanders

Close a diagram by joining each top point to the bottom point beneath it and count the loops: that's the Markov trace, tr(D) = δ^(#loops). It is how the Jones polynomial of a knot is extracted from a braid pushed into TL. And pairing two arch systems — one reflected — and counting the loops of their union is exactly an entry of the meander matrix, whose determinant Di Francesco factored in closed form even though the meander numbers have none. Same algebra, both pages.

History

Why it sits in the geometry pack

It's the algebraic engine of its sibling meander page, and it shares the loop-gas/CFT machinery that links the pack's combinatorics to physics — the way Viazovska's modular forms do for sphere packing. And the wrap question reaches across the site to the toroidal games: the same donut, now carrying loops instead of a maze.

sources