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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 whose diagrams agree outside a small disk.\nInside it, they have a positive crossing, a negative crossing, and the oriented resolution.\nThe *HOMFLY polynomial* is the unique map $P \\colon \\mathcal{L} \\to \\Z[a, a^{-1}, z, z^{-1}]$ such that:\n\n1. $P(O) = 1$ for the [[unknot|1TQ06R4Q66]] $O$;\n2. $a \\, P(L_+) - a^{-1} P(L_-) = z \\, P(L_0)$ for every such triple.\n\n## Remarks\n\nSeveral conventions for the variables are in use; they differ by substitutions.\nExample: a one-crossing diagram of the unknot gives $a - a^{-1} = z \\, P(O \\sqcup O)$, so the two-component unlink has $P = (a - a^{-1}) z^{-1}$.\nNon-example: the map $L \\mapsto 1$ satisfies clause 1 but violates clause 2, since $a - a^{-1} \\ne z$.\nThe substitution $a = t^{-1}$, $z = t^{1/2} - t^{-1/2}$ gives the [[Jones polynomial|4SPC17FPXE]].\nThe substitution $a = 1$, $z = t^{1/2} - t^{-1/2}$ gives the Conway-normalized [[Alexander polynomial|JKA25KDPQP]].\n",
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