When approximating a space curve, it is natural to consider whether the knot type of the original curve is preserved in the approximant. This preservation is of strong contemporary interest in computer graphics and visualization. We establish a criterion to preserve knot type under approximation that relies upon pointwise convergence and convergence in total curvature.
There exist results in the literature showing ambient isotopy from a different point of view @cite_17 @cite_18 . Precisely, there is an upper bound on distance and an upper bound on angles between corresponding points for two curves. If the corresponding distances and angles are within the upper bounds, then they are ambient isotopic. Milnor @cite_15 defined the total curvature for a @math curve using inscribed PL curves. The extension of the definition to piecewise @math curves can be trivially done. Consequently, Fenchel's Theorem can be applied to piecewise @math curves, as we need here.
@cite_18 @cite_15 @cite_17
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2'n, equality holding only for plane convex curves. K. Borsuk, in 1947, extended this result to n dimensional space, and, in the same paper, conjectured that the total curvature of a knot in three dimensional space must exceed 47r. A proof of this conjecture is presented below.' In proving this proposition, use will be made of a definition, suggested by R. H. Fox, of total curvature which is applicable to any closed curve. This general definition is validated by showing that the generalized total curvature K(C) is Generalizing Milnor’s result that an FTC (finite total curvature) knot has an isotopic inscribed polygon, we show that any two nearby knotted FTC graphs are isotopic by a small isotopy. We also show how to obtain sharper constants when the starting curve is smooth. We apply our main theorem to prove a limiting result for essential subarcs of a knot.