Braids and the Braid Group - Examples and Constructions

Explicit braid constructions demonstrate the power of algebraic knot representations and provide computational tools.

Example

Torus knots as braids: The $(p,q)$ -torus knot $T(p,q)$ has canonical braid representation in $B_q$ : $\beta_{p,q} = (\sigma_1 \sigma_2 \cdots \sigma_{q-1})^p$

For trefoil $T(2,3)$ : $\beta = (\sigma_1 \sigma_2)^2 = \sigma_1 \sigma_2 \sigma_1 \sigma_2$

Closure $\overline{\beta}$ gives the trefoil. Different factorizations (e.g., $\sigma_2 \sigma_1 \sigma_2 \sigma_1$ ) represent the same knot via Markov equivalence.

For $T(3,2)$ : $\beta = (\sigma_1)^3 = \sigma_1^3$ (3-twist in 2-strand braid)

Definition

Braid index $b(K)$ of a knot $K$ is the minimum number of strands needed in any braid representation: $b(K) = \min\{n : K = \overline{\beta} \text{ for some } \beta \in B_n\}$

Properties:

$b(\text{unknot}) = 1$
$b(T(p,q)) = \max(p,q)$ for torus knots
$b(K_1 \# K_2) \leq b(K_1) + b(K_2) - 1$
Morton-Franks-Williams inequality: $b(K) \geq s(V_K) + 1$ where $s$ is span of Jones polynomial

Example

Computing braid index: For the figure-eight knot $4_1$ :

Minimal braid: $\beta = \sigma_1 \sigma_2^{-1} \sigma_1 \sigma_2^{-1} \in B_3$
Therefore $b(4_1) = 3$
MFW inequality: $\text{span}(V_{4_1}) = 4$ , giving $b(4_1) \geq 3$
Combined: $b(4_1) = 3$ exactly

For pretzel $P(a,b,c)$ : typically $b \geq 3$ , with equality for many examples.

Remark

Computing braid index exactly is NP-hard in general. Practical approaches:

Upper bound: Convert knot diagram to braid using standard algorithm
Lower bound: Use Jones polynomial span (MFW inequality)
Exact computation: For small knots, enumerate braids in $B_2, B_3, \ldots$ until closure matches

Braid index connects crossing number and bridge number: $c(K) \geq 2b(K) - 2$ for any knot.

Definition

Burau representation: Linear representation $\rho: B_n \to GL_n(\mathbb{Z}[t^{\pm 1}])$ given by: $\rho(\sigma_i) = \begin{pmatrix} I_{i-1} & 0 & 0 \\ 0 & \begin{pmatrix} 1-t & t \\ 1 & 0 \end{pmatrix} & 0 \\ 0 & 0 & I_{n-i-1} \end{pmatrix}$

At $t=-1$ , this gives Alexander polynomial. Burau is faithful for $n \leq 3$ but not for $n \geq 5$ (Moody, 1991).

Example

Plat vs. trace closure: Different closure methods produce different links from the same braid.

For $\beta = \sigma_1 \in B_2$ :

Plat closure: Hopf link (two linked unknots)
Trace closure: Single unknot (isotopically)

The difference matters for link invariants but not for knot theory (single-component links only).

Example

Random braids: Random walks in $B_n$ produce complex knots. Statistical properties:

Typical $n$ -letter random braid in $B_m$ produces knot with $c(K) \approx cn^{1/2}$ crossings
Knot complexity grows sublinearly with braid word length
Most random braids yield hyperbolic knots (Thurston)

Applications: generating test cases for knot algorithms, studying "typical" knot properties.

Definition

Link between braids and mapping class groups: The braid group $B_n$ embeds into the mapping class group $MCG(D_n)$ of an $n$ -punctured disk: $B_n \hookrightarrow MCG(D_n)$

Each braid $\beta$ induces a homeomorphism of the disk fixing boundary, giving a mapping class. This connects braid theory to surface topology and Teichmüller theory.

Example

Quantum groups and R-matrices: The Yang-Baxter equation satisfied by braids corresponds to R-matrices in quantum groups: $R_{12} R_{13} R_{23} = R_{23} R_{13} R_{12}$

Representations of quantum $\mathfrak{sl}_n$ give solutions, producing:

HOMFLY-PT polynomial from $\mathfrak{sl}_n$
Kauffman polynomial from $\mathfrak{so}_n$
Colored Jones polynomials from quantum $\mathfrak{sl}_2$

This quantum group perspective unifies various knot polynomials through representation theory.

These constructions demonstrate how braids provide a computational framework for knot theory, enabling explicit calculations and algorithmic approaches to topological questions.