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  "body": "\n## Summary\n\nThe Thurston norm is a [[seminorm|FP5GVCN8GV]] on the second homology of a compact orientable [[3-manifold|R20TB2KEQQ]].\nIt measures the least complexity, in terms of [[Euler characteristic|RNDDWDAFJ5]], of an embedded surface representing a given class.\nThurston showed that its unit ball is a polyhedron.\nThe classes represented by fibers of fibrations over the circle fill the cones over certain top-dimensional faces.\nIt connects the topology of surfaces in $3$-manifolds with fibrations over the circle.\n\n## Explanations\n\n### L1 human\n\nEvery $2$-dimensional homology class of a $3$-manifold is represented by embedded surfaces, often in many ways.\nThe Thurston norm of the class is the smallest complexity of such a surface, where a surface of genus $g \\ge 1$ counts $2g - 2$ and spheres and disks count nothing.\nFor the complement of a nontrivial knot, the norm of the generator is $2g - 1$ with $g$ the genus of the knot, so the norm detects the simplest Seifert surface.\n",
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