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Abstract

在這篇文章中，我們考慮Normal Projection Energy的一些性質。首先，在$C^{1,1}$平滑性下的Knot，有上界之Normal Projection Energy給出Knot的Gromov's distortion下界。接著，Normal Projection Energy可由total curvature和ropelength之乘積涵蓋住。最後，為求Normal Projection Energy的涵蓋界，我們考慮一類包含在球中並給定端點和總長之曲線的total curvature。

In these paper, we consider several properties of Normal Projection Energy. Firstly, among the class of $C^{1,1}$-smooth knots, the upper bound of Normal Projection Energy gives a uniform lower bound of Gromov's distorsion of knots. Secondly, Normal Projection Energy is bounded by the product of total curvature and ropelength. Thirdly, to prove the bound of Normal Projection Energy, we study the curves which attain the infimum of the total absolute curvature in the set of curves contained in a ball with fixed endpoints and length.