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Braid ​

A braid in Artin's presentation: strands and a word whose letter means and means . Closing it up names a link — and by Alexander's theorem, every link.

WikipediaBraid groupMathWorldBraidGroupWikidataQ220409nLabbraid 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
TorusBraid(p, q), whose closure is

Domain: Braids and knots

Details
  • Relations: for , and
  • Adding gives the symmetric group, which is why surjects onto
  • 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 -cycle
  • An word from the modular group is accepted anywhere a braid is, via its Lorenz braid

Examples

See also: AlexanderPolynomial, LorenzBraid, SeifertGenus