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  "body": "\n## Summary\n\nA ribbon Hopf algebra is a [[quasitriangular Hopf algebra|NCRPWS9W5R]] with a central invertible element, the ribbon element, that behaves like a full twist.\nIts finite-dimensional modules form a ribbon category: a [[braided monoidal category|VT8SFWT0XW]] with duals and compatible twists.\nIn such a category every framed knot or link diagram coloured by modules gives a well-defined morphism.\nSo each ribbon Hopf algebra gives invariants of knots and links.\nSuitable versions of the [[quantum enveloping algebras|1V8H34F6N9]] are ribbon, and the [[Jones polynomial|4SPC17FPXE]] arises this way from $U_q(\\mathfrak{sl}_2)$.\n\n## Explanations\n\n### L1 human\n\nThink of a ribbon, a flat band, rather than a thin string.\nA ribbon can be twisted by a full turn, and that twist cannot be undone by sliding the band around.\nIn a ribbon Hopf algebra a special element $v$ records this full twist.\nThe braiding from the R-matrix records how two ribbons cross, and $v$ records how one ribbon twists.\nTogether they give exactly the bookkeeping needed to turn pictures of tangled bands into algebra.\n",
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