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  "body": "\n## Definition\n\nLet $M$ be a [[$3$-manifold|R20TB2KEQQ]] and let $\\xi$ be a contact structure on $M$, so that $(M, \\xi)$ is a [[contact manifold|HBQ8BZ7XB5]].\nAn *overtwisted disc* in $(M, \\xi)$ is an embedded closed disc $D \\subset M$ such that:\n\n1. the boundary $\\partial D$ is a [[Legendrian|Q2DMSTWZYZ]] curve, that is, tangent to $\\xi$;\n2. for every $x \\in \\partial D$, the plane $\\xi_x$ equals the tangent plane $T_x D$.\n\nThe contact structure $\\xi$ is *overtwisted* if $(M, \\xi)$ contains an overtwisted disc, and *tight* otherwise.\n\n## Remarks\n\nExample: on $\\R^3$ with cylindrical coordinates $(r, \\theta, z)$, the form $\\alpha = \\cos r\\, dz + r \\sin r\\, d\\theta$ is a contact form.\nThe disc $D = \\set{z = 0, \\ r \\le \\pi}$ is overtwisted: on its boundary $r = \\pi$ one has $\\alpha = -dz$, so $\\xi = \\ker dz$ is the horizontal plane, which is $T_x D$, and the boundary circle is tangent to it.\nSo this contact structure is overtwisted.\n\nNon-example: the standard contact structure $\\ker(dz - y\\,dx)$ on $\\R^3$ is tight, by Bennequin's theorem, so no disc satisfies both clauses.\n\nClause 2 says that the framing of $\\partial D$ given by the contact planes agrees with the framing given by $D$.\nIn other words, the Legendrian unknot $\\partial D$ has Thurston–Bennequin number $0$.\nEliashberg proved that two overtwisted contact structures on a closed $3$-manifold are isotopic if and only if they are homotopic as plane fields.\nHe also proved that every homotopy class of plane fields contains an overtwisted contact structure.\nTight contact structures are much rarer; for example, $S^3$ has a unique tight contact structure up to isotopy, also by Eliashberg.\n",
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