REFERENCE SHEET

Braid groups & closures

Record strand motion while retaining information that a final permutation forgets.

Before this Groups & subgroups, Knots, links & invariants

σᵢσᵢ₊₁σᵢ = σᵢ₊₁σᵢσᵢ₊₁; σᵢσⱼ=σⱼσᵢ if |i−j|≥2

Definitions

Braid group Bₙ

Isotopy classes of n strands monotone in height, joining fixed top endpoints to the same set of bottom endpoints. Multiplication stacks braids; inverses reverse crossings and stacking order.

Artin generator

σᵢ exchanges adjacent strands i and i+1 by a specified positive crossing; σᵢ⁻¹ uses the opposite crossing.

Closure

Connect corresponding top and bottom endpoints outside the braid to obtain a link. The endpoint permutation determines the number of closure components via its cycle count.

Key theorems

Artin presentation

Bₙ is generated by σ₁,…,σₙ₋₁ with the two displayed families of relations. Unlike in Sₙ, there is no relation σᵢ²=e.

Alexander theorem

Every oriented tame link is isotopic to the closure of some braid.

Markov theorem

Two braid closures represent the same oriented link type exactly when the braids are connected by conjugation and stabilization or destabilization β↔βσₙ±¹, allowing a change in strand count.

Notable applications

Algebraic knot descriptions

Use braid words to describe links and manipulate them algebraically.

Motion planning

Braids record collision-free exchanges of points in a plane, including winding missed by a permutation alone.

Work it out

Try before revealing

01 / Compute

In B₂, does σ₁² equal the identity? What is its endpoint permutation?

Hint

B₂ has one generator and no defining relation imposing finite order.

Solution

No. B₂≅ℤ, with σ₁ corresponding to 1, so σ₁² corresponds to 2. Its endpoint permutation is the identity because the two exchanges cancel in S₂.

02 / Compute

How many components does the closure of a braid with endpoint permutation (123)(45) have?