Braids and the Braid Group - Core Definitions

Braids provide an algebraic framework for studying knots through group theory, connecting topology to algebra via explicit presentations.

Definition

A braid on $n$ strands is a collection of $n$ non-intersecting strands connecting $n$ points on a top line to $n$ points on a bottom line, monotonically decreasing in the vertical direction. Two braids are equivalent if one can be continuously deformed to the other keeping endpoints fixed.

The braid group $B_n$ is the set of equivalence classes of $n$ -strand braids under concatenation (stacking braids vertically).

Unlike knots (which are embedded circles), braids have distinct "top" and "bottom," making them easier to represent algebraically. Every knot can be represented as the closure of some braid, connecting braid theory to knot theory.

Definition

The Artin generators $\sigma_1, \ldots, \sigma_{n-1}$ of $B_n$ represent elementary crossing braids:

$\sigma_i$ : strand $i$ crosses over strand $i+1$ , all other strands descend straight
$\sigma_i^{-1}$ : strand $i$ crosses under strand $i+1$

The braid group has presentation: $B_n = \langle \sigma_1, \ldots, \sigma_{n-1} \mid \sigma_i \sigma_j = \sigma_j \sigma_i \text{ for } |i-j| \geq 2, \, \sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1} \rangle$

The second relation is the Yang-Baxter equation or braid relation.

Example

3-strand braid group $B_3$ : $B_3 = \langle \sigma_1, \sigma_2 \mid \sigma_1 \sigma_2 \sigma_1 = \sigma_2 \sigma_1 \sigma_2 \rangle$

Elements include:

Identity: $e$ (all strands descend straight)
$\sigma_1$ : left strand crosses over middle
$\sigma_1 \sigma_2 \sigma_1$ : makes a trefoil when closed
$\sigma_1^{-1} \sigma_2^{-1} \sigma_1^{-1}$ : makes mirror trefoil

The center of $B_3$ is infinite cyclic: $Z(B_3) = \langle \Delta^2 \rangle$ where $\Delta = \sigma_1 \sigma_2 \sigma_1$ is the Garside element.

Remark

The braid relation $\sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}$ appears throughout mathematics:

Yang-Baxter equation in statistical mechanics
Quantum groups and their representations
Hecke algebras in representation theory
Knot polynomials via braid representations

This universality makes braid groups central to modern mathematical physics.

Definition

The closure of a braid $\beta \in B_n$ creates a link by connecting top endpoints to bottom endpoints:

Plat closure: connect top $i$ to bottom $i$ directly
Trace closure: connect cyclically (top $i$ to bottom $i+1 \pmod{n}$ )

Every link arises as the closure of some braid (Alexander's theorem, 1923). Two braids have equivalent closures if and only if they are related by Markov moves:

Conjugation: $\beta \mapsto \gamma \beta \gamma^{-1}$ in $B_n$
Stabilization: $\beta \in B_n \mapsto \beta \sigma_n^{\pm 1} \in B_{n+1}$

Example

The trefoil knot from braids:

Right-handed: $\overline{\sigma_1 \sigma_2 \sigma_1} = \overline{\sigma_2 \sigma_1 \sigma_2}$ (braid relation makes these equal)
Left-handed: $\overline{\sigma_1^{-1} \sigma_2^{-1} \sigma_1^{-1}}$
Figure-eight: $\overline{\sigma_1 \sigma_2^{-1} \sigma_1 \sigma_2^{-1}} \in B_3$

Different braid words can represent the same knot, resolved by Markov's theorem.

The braid group perspective transforms knot theory into group theory, enabling algebraic techniques (representation theory, homological algebra) to attack topological problems.