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  "body": "\n## Summary\n\nContact structures on $3$-manifolds split into two classes.\nAn overtwisted contact structure contains an overtwisted disc, a disc whose boundary circle is tangent to the contact planes in a special twisted way.\nA tight one contains no such disc.\nEliashberg showed that overtwisted structures are classified by homotopy theory alone, so they are flexible.\nTight structures are rigid and carry the deep information of three-dimensional [[contact topology|HBQ8BZ7XB5]].\n\n## Explanations\n\n### L1 human\n\nA contact structure is a field of planes on a three-dimensional space that twists everywhere.\nIn an overtwisted one, the planes twist so much that one can find a special disc.\nIts rim runs along the planes, and all along the rim the planes lie flat on the disc.\nA tight contact structure never twists that much.\nThe standard structure on $\\R^3$ is tight, by a theorem of Bennequin, while for example the planes $\\ker(\\cos r\\, dz + r \\sin r\\, d\\theta)$ in cylindrical coordinates are overtwisted.\n",
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