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  "body": "\n## Definition\n\nLet $S$ be a compact connected orientable [[surface|VGJJVBZ1MQ]] of [[genus|NJ787GZS1N]] $g$ with $n$ boundary components, and assume $3g - 3 + n \\ge 2$.\nA simple closed curve in $S$ is *essential* when it bounds no disc in $S$.\nIt must also not cobound an annulus with a component of $\\partial S$.\nThe *curve complex* $\\mathcal{C}(S)$ is the [[simplicial complex|V8GZ0W1ZZ5]] such that:\n\n1. its vertices are the isotopy classes of essential simple closed curves in $S$;\n2. a set of $k + 1$ distinct vertices spans a $k$-simplex if and only if the classes have pairwise disjoint representatives.\n\n## Remarks\n\nA maximal set of pairwise disjoint, pairwise non-isotopic essential curves has $3g - 3 + n$ elements, a pants decomposition, so $\\mathcal{C}(S)$ has dimension $3g - 4 + n$.\nThe [[mapping class group|JV5NYG913V]] acts on $\\mathcal{C}(S)$, and under the hypothesis $3g - 3 + n \\ge 2$ the complex is connected and has infinite diameter.\n\nExample: for the sphere with $5$ boundary circles, $g = 0$ and $n = 5$, so $3g - 3 + n = 2$ and $\\mathcal{C}(S)$ is a graph.\nEach essential curve cuts off two of the boundary circles from the other three, and two such curves are joined by an edge when they can be made disjoint.\n\nNon-example of clause 2: on a closed surface of genus $2$, let curves $a$ and $b$ meet transversally in one point.\nTheir algebraic intersection number is $\\pm 1$, so they have no disjoint representatives.\nSo $a$ and $b$ are not joined by an edge.\n\nFor the torus, the once-punctured torus and the four-holed sphere, the definition gives a complex without edges.\nFor them one joins curves of minimal intersection by edges instead, and the result is the Farey graph.\n\n## Explanations\n\n### L1 human\n\nThe vertices are loops on the surface that cannot be shrunk away, counted up to sliding.\nSimplices record which loops can coexist without crossings.\nThe condition on $g$ and $n$ excludes small surfaces where no two different loops are disjoint.\n",
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