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  "body": "\n## Statement\n\nLet $\\gamma \\colon [a, b] \\to \\R^3$ be a regular simple [[closed curve|0D032NSZKY]] of class $C^2$ whose derivatives up to order $2$ also agree at $a$ and $b$.\nIf the [[total curvature|QP9DY5K8QW]] of $\\gamma$ is at most $4\\pi$, then $\\gamma$ is unknotted, that is, its image is a trivial [[knot|QYEAZTYGPT]].\nEquivalently, every knotted curve of this kind has total curvature greater than $4\\pi$.\n\n## Remarks\n\nMilnor's proof uses height functions.\nFor almost every unit vector $v$, the function $t \\mapsto \\ip{\\gamma(t)}{v}$ has finitely many local maxima, and the total curvature equals $2\\pi$ times the average number of these maxima over all $v$.\nIf the total curvature is less than $4\\pi$, some direction $v$ gives exactly one local maximum and one local minimum.\nThen each plane orthogonal to $v$ meets the curve in at most two points, and joining these pairs of points gives an embedded disc bounded by the curve.\nSo the curve is unknotted; the case of total curvature exactly $4\\pi$ needs a further argument.\nExample: a standard trefoil knot must have total curvature greater than $4\\pi$, while a round circle, with $2\\pi$, is unknotted.\nIstván Fáry and John Milnor proved the theorem independently, around 1949 and 1950.\n\n## Explanations\n\n### L1 human\n\nA knotted loop of wire must bend a lot.\nIn total, it must turn by more than two full turns.\nSo if a closed loop turns by at most two full turns, it can be pulled into a round circle without cutting.\n\n### L2 human\n\nThe theorem refines [[Fenchel's theorem|F0BJ2BHSZG]], which gives the bound $2\\pi$ for every closed curve.\nThe bound $4\\pi$ is sharp: there are knotted curves with total curvature as close to $4\\pi$ as one likes.\nMilnor's averaging formula for the total curvature is a model of later integral-geometric methods, such as the Crofton formula.\n",
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