A lower bound of the total curvature of a knotted curve in R^n
Reeves, Amelia L.
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In 1929, Fenchel proved that a closed plane curve must have total curvature, 2¼, with equality holding for convex curves. Borsuk showed later in 1947 that the same is true for curves in R^n. He conjectured that for nontrivial knots in 3 dimensions, the total curvature must be at least 4¼. István Fáry proved this conjecture in 1949 in three main theorems. Fáry first proved that the total curvature of a space curve is equal to the limit of the total curvature of a sequence of polygons inscribed within the curve. He then showed that the total curvature of a space polygon is the average of the total curvatures of its orthogonal projections onto planes. Last, Fáry observed that the total curvature of a planar projection of a knotted curve is at least 4¼. This is an exposition of Fáry's proof.