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  "body": "\n## Definition\n\nLet $K$ be a tame [[knot|QYEAZTYGPT]] in $\\R^3$.\nA *diagram* of $K$ is a projection of $K$ to a plane that is injective except at finitely many transverse double points, called *crossings*.\nAt each crossing the diagram records which strand passes over.\nThe *crossing number* $c(K)$ is the least number of crossings of a diagram of a knot equivalent to $K$.\n\n## Remarks\n\nSince the minimum is taken over all knots equivalent to $K$, the number $c(K)$ is a [[knot invariant|GZQ5ZE3X6G]].\nExample: $c(\\text{unknot}) = 0$, since the round circle has a diagram with no crossings.\nFor the trefoil, $c = 3$: it has a diagram with $3$ crossings, and every diagram with at most $2$ crossings represents the unknot, which is not equivalent to the trefoil.\nNon-example: a diagram of the unknot with one twist has $1$ crossing, but $1$ is not the crossing number of the unknot, because the minimum is $0$.\nFor a reduced alternating diagram, the number of crossings equals the crossing number; this was one of the Tait conjectures, proved with the Jones polynomial.\nThe same definition applies to links.\n",
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  "name": "Crossing number",
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