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  "body": "\n## Statement\n\nLet $L$ be a tame oriented [[link|1A5WRCNTM3]] in $\\R^3$.\nThen there are an integer $n \\ge 1$ and a braid $\\beta$ in the [[braid group|BNR4H5GCTP]] $B_n$ with this property.\nThe link $L$ is ambient isotopic, as an oriented link, to the closure $\\hat\\beta$ of $\\beta$.\nHere $\\hat\\beta$ is obtained from a picture of $\\beta$ in a cylinder in the following way.\nThe top end of each strand is joined to the bottom end in the same position, outside the cylinder and without new crossings.\nAll strands are oriented downward.\n\n## Remarks\n\nThe idea of the proof is to choose an axis and to make the link wind around it in one direction.\nTake a polygonal diagram of $L$ and a point $O$ of the plane off the diagram.\nEach edge that runs clockwise around $O$ can be replaced by two edges that pass on the other side of $O$, so that they run counterclockwise.\nAfter finitely many such moves every edge runs counterclockwise around $O$, and cutting along a ray from $O$ shows the link as a closed braid.\nFor example, the trefoil knot is the closure of $\\sigma_1^3$ in $B_2$, and the Hopf link is the closure of $\\sigma_1^2$ in $B_2$.\nThe braid $\\beta$ is far from unique; the [[Markov theorem|7F3QQQ63WR]] says exactly when two braids have the same closure.\n\n## Explanations\n\n### L1 human\n\nEvery knot or link can be drawn as a braid whose ends are joined around the back.\nImagine a pole through the middle of the knot: you can move the string so that it always goes around the pole in the same direction.\nThen the string looks like a braid that has been wrapped into a ring.\n\n### L2 human\n\nThe theorem turns questions about links into questions about the braid groups $B_n$, which are groups with simple presentations.\nTogether with the [[Markov theorem|7F3QQQ63WR]], it shows that a link invariant is the same as a function on all braid groups that is unchanged by the two Markov moves.\nThis is how Jones first found the [[Jones polynomial|4SPC17FPXE]], from traces on representations of braid groups.\nThe least $n$ for which $L$ is a closed $n$-braid is called the braid index of $L$.\n",
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  "name": "Alexander's theorem on closed braids",
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