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  "body": "\n## Statement\n\nFor $n \\ge 1$, let $B_n$ be the [[braid group|BNR4H5GCTP]] with generators $\\sigma_1, \\ldots, \\sigma_{n-1}$.\nView $B_n$ as a subgroup of $B_{n+1}$ by adding a straight strand on the right.\nFor a braid $\\beta$, let $\\hat\\beta$ be its closure, an oriented [[link|1A5WRCNTM3]] in $\\R^3$.\nThe two Markov moves on braids are:\n\n1. Conjugation: replace $\\beta \\in B_n$ by $\\gamma \\beta \\gamma^{-1}$ for some $\\gamma \\in B_n$, or the converse.\n2. Stabilization: replace $\\beta \\in B_n$ by $\\beta \\sigma_n$ or by $\\beta \\sigma_n^{-1}$ in $B_{n+1}$, or the converse.\n\nLet $\\beta \\in B_n$ and $\\beta' \\in B_m$.\nThen $\\hat\\beta$ and $\\hat\\beta'$ are ambient isotopic as oriented links if and only if a finite sequence of Markov moves turns $\\beta$ into $\\beta'$.\n\n## Remarks\n\nOne direction is a direct check.\nClosing up $\\gamma \\beta \\gamma^{-1}$ lets $\\gamma$ slide around the closure and cancel $\\gamma^{-1}$, so the closure does not change.\nClosing up $\\beta \\sigma_n^{\\pm 1}$ adds one strand with a single kink, which a first [[Reidemeister move|T3HAGWYKK7]] removes.\nThe hard direction follows an isotopy of closed braids step by step and shows that each step is a combination of Markov moves.\nFor example, $\\sigma_1 \\in B_2$ and the identity braid in $B_1$ both close up to the unknot, and they differ by one stabilization.\nConjugation alone does not suffice, since conjugation never changes the number of strands, while braids with different numbers of strands can close to the same link.\n\n## Explanations\n\n### L1 human\n\nMany different braids close up to the same knot.\nThe theorem lists two simple changes, and says these are the only ones you ever need.\nYou may slide a piece of the braid around the closure to the other end, and you may add or remove an extra strand with one twist.\n\n### L2 human\n\nBy [[Alexander's theorem|PRWP1JF486]], every oriented link is a closed braid.\nThe Markov theorem then says that oriented links up to isotopy are the same as braids in all $B_n$ up to the two Markov moves.\nSo a family of functions $f_n$ on $B_n$ gives a link invariant when $f_n$ is constant on [[conjugacy classes|0DEBXAJ5SA]] and $f_{n+1}(\\beta \\sigma_n^{\\pm 1})$ equals $f_n(\\beta)$.\nTraces on the Temperley–Lieb and Hecke algebras with this property give the [[Jones polynomial|4SPC17FPXE]] and the [[HOMFLY polynomial|QBQ0BH589C]].\n",
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