Teaching objectives
A group theory lab for first-year university students. Braids are a group you can touch: you weave them from crossings, add them by stacking one under the other and undo them with their mirror image. The group is infinite, non-abelian and has a presentation with only two kinds of relations, so it revisits almost everything in the course (generators, presentations, homomorphisms, kernels, quotients, conjugation, centre, abelianisation) with a fresh example. At the end, closing the braids produces knots, and the invariants that tell them apart turn out to be homomorphisms.
Each room has between four and eight challenges, one after another. The strands have fixed neon colours, so you can see where each one ends without explaining it.
What students learn
- Read and write braids as words in the generators σᵢ and σᵢ⁻¹, and go from the picture to the word and back.
- The group operation: adding two braids by stacking one under the other, the identity, the inverse as a mirror image, and the fact that it is not commutative.
- Artin's presentation: far-apart crossings commute, and the braid relation σᵢσᵢ₊₁σᵢ = σᵢ₊₁σᵢσᵢ₊₁. Define homomorphisms by checking the relations.
- Simplify words with these moves and decide whether a braid is trivial (the word problem), using Artin's action on the free group and Garside's normal form.
- The structure of Bₙ: the homomorphism onto Sₙ and its kernel (the pure braids), the abelianisation ℤ (counting signed crossings) and the centre generated by the full twist Δ².
- Close a braid to get a knot or a link, count its components, untangle it with Reidemeister moves and recognise when two braids give the same link (Markov).
- Tell knots apart with invariants: the linking number, tricolourings and the Jones polynomial.
Key mathematical ideas
- Three moves are enough. Every deformation of a braid is a chain of three kinds of steps: undoing a pair σσ⁻¹, commuting far-apart crossings and the braid relation (Artin's theorem).
- There are infinitely many braids, but they can be compared. The action on the free group is faithful, and Garside's normal form gives a unique way of writing each braid: this is how a computer decides whether two words are the same braid.
- Forgetting which strand is on top is a homomorphism. π: Bₙ → Sₙ is onto; its kernel Pₙ is infinite, so Bₙ is much bigger than Sₙ.
- All the σᵢ are conjugate, so any homomorphism to an abelian group sends them all to the same element and the abelianisation is ℤ.
- An invariant is something Reidemeister moves do not change. Tricolourings are homomorphisms from the knot group to the dihedral group Δ₃. The trefoil has 9 and the unknot 3, so they are not the same knot.
Sources and credits
The presentation follows E. Artin, “Theorie der Zöpfe” (1925) and “Theory of braids” (1947). The normal form is F. A. Garside's (1969). For the whole lab, C. Kassel and V. Turaev, Braid Groups (Springer, GTM 247, 2008). Also the theorems of Alexander (1923) and Markov (1935), R. H. Fox's colourings, and L. H. Kauffman's bracket (1987) for the polynomial of V. F. R. Jones (1985).
Classroom notes
- “The same braid?” works very well on a projector: let the class vote before watching the animated deformation.
- “The braid adder” is a good moment to review the group axioms with an object you can touch (some string or shoelaces on the table help).
- “The full twist” is easier to understand with the belt trick: give one end of a belt a full turn and see the strands wind around each other without really crossing.
- “Three colours” is the first proof that a knot cannot be undone: it deserves a pause.
Room-by-room contents
Room 1 · Weave your first braid
What the braid group is used for today (topological quantum computing, cryptography, DNA) and some history (Hurwitz, Artin). Definition of a braid and of the crossings σᵢ and σᵢ⁻¹. Four braids are copied with a crossing keypad, and it turns out there are infinitely many.
Student tasks
- Copy four drawn braids.
- Decide how many braids on 3 strands there are.
Room 2 · The same braid?
Six pairs of braids. If they are the same, the animated deformation is shown (commuting, undoing a pair, braid relation); if not, the invariant that prevents it: where each colour ends or the exponent sum.
Student tasks
- Decide for six pairs whether they are the same braid.
Room 3 · From word to picture
From word to picture (thumbnails with distractors: inverse, reversed order, mirror, one sign changed) and from picture to word.
Student tasks
- Choose the picture of three words.
- Write the word of three pictures.
Room 4 · Where does each colour end up?
The homomorphism π: Bₙ → Sₙ. Where a strand ends, the permutation in cycles, π(ab) = π(a)π(b), that it is onto and that its kernel (σ₁²) consists of the pure braids.
Student tasks
- Follow two strands.
- Write a permutation in cycles.
- Compose two braid permutations.
- Find a braid in the kernel.
Room 5 · The braid adder
The adder stacks one braid under another with an animation. Review of the group axioms: which picture is A·B, whether A·B = B·A, the identity, associativity and an equation σ₁σ₂·B = σ₂σ₁σ₂.
Student tasks
- Recognise two sums.
- Decide whether adding is commutative.
- Solve an equation in B₃.
Room 6 · The braid that undoes
The inverse is the top-to-bottom mirror; adding A and A⁻¹ untangles like a zip. (AB)⁻¹ = B⁻¹A⁻¹ and the infinite order of σ₁.
Student tasks
- Write three inverses.
- Choose (AB)⁻¹.
- Decide the order of σ₁.
Room 7 · Far-apart crossings
σᵢσⱼ = σⱼσᵢ when |i − j| ≥ 2. Puzzles moving far-apart crossings to a target word and questions about which generators commute.
Student tasks
- Solve two commuting puzzles.
- Decide three commutations.
- See that ⟨σ₁, σ₃⟩ is abelian.
Room 8 · The braid relation
σᵢσᵢ₊₁σᵢ = σᵢ₊₁σᵢσᵢ₊₁ animated, its variants with inverses, and conjugation: all the σᵢ are conjugate.
Student tasks
- Solve two puzzles with R3.
- Find another word for σ₂σ₃σ₂.
- Find c with cσ₁c⁻¹ = σ₂.
Room 9 · Braids side by side
Putting A and B side by side: Bₘ × Bₙ → Bₘ₊ₙ, shifting the indices of B. It is an injective homomorphism.
Student tasks
- Write two words of A ⊕ B.
- Recognise a picture.
- Decide three properties.
Room 10 · Artin’s presentation
Artin’s theorem and how a homomorphism is defined from a presentation by checking the relations, into S₃, S₄ and ℤ.
Student tasks
- Count the relations of B₄ and B₅.
- Decide whether four assignments define homomorphisms.
- Recognise B₂.
Room 11 · Undoing pairs
Why deciding whether two braids are equal matters (the word problem). Puzzles with R2 only, and the uniqueness of free reduction.
Student tasks
- Solve four R2 puzzles.
- Decide whether the order matters.
- Compute two reduced lengths.
Room 12 · The simplifier
Five puzzles with R2, commuting and R3 down to the shortest word, compared with the simplifier’s record, which can be watched solving them.
Student tasks
- Solve five simplification puzzles.
Room 13 · Is the braid trivial?
Artin’s action on the free group with a calculator that updates the images of x₁, …, xₙ. Since the action is faithful, it decides whether a braid is trivial.
Student tasks
- Compute two images.
- Decide whether three braids are trivial.
- Understand what a faithful action means.
Room 14 · Half twist and normal form
The half twist Δ, simple braids and Garside’s normal form Δᵏ·A₁⋯A_r drawn in boxes.
Student tasks
- Recognise simple braids.
- Count the simple braids of B₃ and B₄.
- Compute |Δ₅| and Δσ₁Δ⁻¹.
- Compare two pairs of words with the normal form.
Room 15 · Pure braids
Pₙ = ker π, normal and of index n!. Generators Aᵢⱼ, P₂ ≅ ℤ and cosets as permutations.
Student tasks
- Recognise pure braids.
- Compute [B₄ : P₄].
- Choose a braid in the same coset.
- Read which strands Aᵢⱼ links.
Room 16 · Counting signed crossings
The exponent sum is a homomorphism to ℤ; since all the σᵢ are conjugate, the abelianisation is ℤ and the commutator subgroup is its kernel.
Student tasks
- Compute two exponent sums.
- Decide about homomorphisms to ℤ and ℤ₆.
- Decide whether σ₁σ₂⁻¹ is a product of commutators.
Room 17 · The full twist
3D scene: turning the bottom disc 360° weaves the full twist Δ². Δ² is central, Δ does not commute with σ₁, and the centre of Bₙ is ⟨Δ²⟩.
Student tasks
- Compute the length and exponent sum of Δ².
- Decide whether Δ² is pure and whether Δ is central.
- Choose the central braid.
Room 18 · Closing the braid
Why knots matter (DNA, proteins, physics) and their history. Closing a braid gives a knot or a link; the components are the cycles of the permutation. Alexander’s theorem.
Student tasks
- Count the components of three closures.
- Choose the one that is a knot.
- Match trefoil, Hopf and figure-eight with their braids.
Room 19 · Untangle the knot
Reidemeister moves on the closed braid, with animated demos. Five knots that untangle with R1, R2, R3, commuting and rotating, and the trefoil, which does not.
Student tasks
- Untangle five diagrams.
- Decide whether the trefoil can be untangled.
Room 20 · Markov moves
Two braids give the same link if and only if one can be turned into the other by conjugating and stabilising. Pairs decided by components and by the Jones polynomial.
Student tasks
- Decide five pairs.
- Stabilise σ₁³ to σ₁³σ₂.
- Recognise rotating as conjugating.
Room 21 · Linking number
lk = half the sum of the signs of the crossings between two components. The Whitehead link has lk = 0 and is still linked.
Student tasks
- Compute four linking numbers.
- Decide whether the Whitehead link is trivial.
- Explain why lk is invariant.
Room 22 · Three colours
Why invariants are needed. Tricolourings as homomorphisms from the knot group to Δ₃: the trefoil and 6₁ can be coloured, the figure-eight cannot; the trefoil has 9 and the unknot 3, so they differ. 5-colourings of the figure-eight.
Student tasks
- Colour the trefoil and 6₁.
- Decide whether the figure-eight can be coloured.
- Count the colourings of the trefoil and the unknot.
Room 23 · The Jones polynomial
The Kauffman bracket with a worked example, the guided computation for the Hopf link, the chirality of the trefoil and a final challenge: recognise three disguised knots with an invariant calculator.
Student tasks
- Compute the bracket of the Hopf link step by step.
- Compare the trefoil with its mirror image.
- Identify three knots.
Rooms to project
The most striking ones to show and discuss in class.