Definition 12.3.1 (Braid group). The braid group on strands has presentation This is the same as the presentation of but without the relations . The generator represents crossing strand over strand .
Note. There is a natural surjection sending each generator to the transposition . The kernel is the pure braid group . Unlike , the braid group is infinite: in .
Braids are best understood as diagrams: strands running from top to bottom, where crosses strand over strand , and crosses strand over strand . Composition is vertical stacking. Two braid diagrams represent the same element iff one can be deformed into the other (isotopy). The braid relation (the Yang–Baxter equation) says these two braids are isotopic:
Note. Markov’s theorem connects braids to knot theory: every knot or link in can be represented as the closure of a braid, and two braids give isotopic links iff they are related by conjugation in and stabilization .
Note. is much larger than , but its algorithmic aspects can be approached by the same methods, using permutation braids (braids where any two strands cross at most once, in bijection with ). Let be the longest permutation braid. Since its shortest word can start/end with any , and is still a permutation braid, every element of can be written as where the are permutation braids satisfying a “left greedy” condition. This is the Garside normal form (Garside 1969, Thurston–ElRifai–Morton early 1990s), which solves the word problem in .