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  "body": "\n## Definition\n\nLet $\\mathcal{L}$ be the set of isotopy classes of oriented tame [[links|1A5WRCNTM3]] in $S^3$.\nLet $L_+$, $L_-$ and $L_0$ be oriented links with diagrams that agree outside a small disk.\nInside the disk, $L_+$ has a positive crossing, $L_-$ has a negative crossing, and $L_0$ has the two strands resolved without crossing, respecting the orientation.\nThe *Jones polynomial* is the unique map $V \\colon \\mathcal{L} \\to \\Z[t^{1/2}, t^{-1/2}]$ such that:\n\n1. $V(O) = 1$ for the [[unknot|1TQ06R4Q66]] $O$;\n2. $t^{-1} V(L_+) - t \\, V(L_-) = (t^{1/2} - t^{-1/2}) \\, V(L_0)$ for every such triple.\n\n## Remarks\n\nExistence is the main theorem; uniqueness holds because changing crossings in any diagram reaches an unlink, and the relation computes the rest.\nExample: a one-crossing diagram of the unknot gives $L_+ = L_- = O$ and $L_0$ the two-component unlink $O \\sqcup O$.\nThe relation gives $t^{-1} - t = (t^{1/2} - t^{-1/2}) V(O \\sqcup O)$, so $V(O \\sqcup O) = -(t^{1/2} + t^{-1/2})$.\nFor the trefoils, $V = t + t^3 - t^4$ for one and $V = t^{-1} + t^{-3} - t^{-4}$ for its mirror image.\nNon-example: the map $L \\mapsto 1$ satisfies clause 1 but not clause 2, since $t^{-1} - t \\ne t^{1/2} - t^{-1/2}$.\nIn terms of the [[Kauffman bracket|JYKVYS4N32]], $V_L(t) = (-A^3)^{-w(D)} \\langle D \\rangle$ at $A = t^{-1/4}$, where $w(D)$ is the writhe of a diagram $D$ of $L$.\nMirror image changes $V_L(t)$ to $V_L(t^{-1})$.\n",
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