Braid

A braid in Artin's presentation: $n$ strands and a word whose letter $k$ means $\sigma_k$ and $-k$ means $\sigma_k^{-1}$. Closing it up names a link — and by Alexander's theorem, every link.

WikipediaBraid group↗MathWorldBraidGroup↗WikidataQ220409↗nLabbraid group↗

Braid(strands, word)the braid

BraidPermutation(braid)the image in the symmetric group — forget which strand went over

BraidWrithe(braid)the exponent sum, i.e. the abelianisation $B_n \to \mathbb{Z}$

TorusBraid(p, q)$(\sigma_1\cdots\sigma_{p-1})^q$, whose closure is $T(p,q)$

Typed signatures

Type	Package	Applies
(integer, list<integer>?) -> expression<Braid>	braid	always

Domain: Braids and knots·Drawn as Graphics

Details

• Relations: $\sigma_i\sigma_j = \sigma_j\sigma_i$ for $|i-j| \ge 2$, and $\sigma_i\sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i\sigma_{i+1}$
• Adding $\sigma_i^2 = 1$ gives the symmetric group, which is why $B_n$ surjects onto $S_n$
• Two different words can name the same braid; nothing here solves the word problem, so only invariants are computed
• BraidComponents counts the permutation's cycles — the closure is a knot exactly when it is an $n$-cycle
• An $LR$ word from the modular group is accepted anywhere a braid is, via its Lorenz braid

Examples

Basic #

σ₁σ₂ cycles the three strands

live ↗

In$\mathrm{BraidPermutation}(\mathrm{Braid}(3, \bigl\lbrack1, 2\bigr\rbrack))$Out$\bigl\lbrack2, 3, 1\bigr\rbrack$

Scope #

two positive crossings and one negative

live ↗

In$\mathrm{BraidWrithe}(\mathrm{Braid}(3, \bigl\lbrack1, -2, 1\bigr\rbrack))$Out$1$

Properties #

the Hopf link

live ↗

In$\mathrm{BraidComponents}(\mathrm{Braid}(2, \bigl\lbrack1, 1\bigr\rbrack))$Out$2$

$T(2,q)$ is a knot for odd $q$

live ↗

In$\mathrm{BraidIsKnot}(\mathrm{TorusBraid}(2, 5))$Out$\mathrm{True}$

About

Examples	4
Crosswalk	WikipediaMathWorldWikidatanLab

See also: AlexanderPolynomial, LorenzBraid, SeifertGenus