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  "body": "\n## Summary\n\nThe braid group $B_n$ is the [[group|G5S1M9]] of braids with $n$ strands.\nTwo braids are equal when one can be deformed into the other, and the product places one braid after the other.\nArtin gave its presentation with generators $\\sigma_1, \\ldots, \\sigma_{n-1}$, the crossings of neighbouring strands, so $B_n$ is an [[Artin group|SAWBDBHE41]].\nIt is also the [[fundamental group|S1XK3HXHFX]] of the space of sets of $n$ distinct points in the plane.\nIt is also the [[mapping class group|JV5NYG913V]] of a disk with $n$ marked points.\nBraid groups link algebra with knot theory: by [[Alexander's theorem|PRWP1JF486]], every link is the closure of a braid.\n\n## Explanations\n\n### L1 human\n\nFix $n$ pegs on a top bar and $n$ pegs on a bottom bar, and join them with $n$ strings that always go downward.\nTwo such braids are the same if one can be moved into the other without cutting the strings or moving the pegs.\nTo multiply two braids, hang one below the other.\nThe basic braid $\\sigma_i$ crosses string $i$ over string $i + 1$, and its inverse crosses it back under.\nFor example, with three strands, $\\sigma_1 \\sigma_2 \\sigma_1$ and $\\sigma_2 \\sigma_1 \\sigma_2$ are the same braid: both reverse the order of the three strings.\n",
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