臺灣博碩士論文加值系統

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.

1. Introduction........................................1
2. Lower bounds for thickness of knots or links as a function of their Normal Projection Energy.............3
3. The total absolute curvature of piecewise $C^{2}$ open curve in $R^{3}$......................................14
4. A upper bound of Normal Projection Energy and
Finiteness of Knot Type...............................18
5. References.........................................26

[1] G. Buck and J. Orloff, A simple energy function for knots, Topology Appl. 61 (1995), no. 3, 205-214.
[2] G. Buck and J. Simon, Thickness and crossing number of knots, Topology Appl. 91 (1999), no. 3, 245-257.
[3] G. Buck and J. Simon, Total curvature and packing of knots, Topology Appl. 154 (2007), no. 1, 192-204.
[4] K. Enomoto, J. I. Itoh and R. Sinlair, The total absolute curvature of open curves in E3, Illinois Journal of Mathematics, Volume 52, No.1, Spring 2008, Pages 47-76.
[5] M. Freeddman, Z. -X. He and Z. Wang, M¨obius energy of knots and unknots, Ann. of Math. 139 (1994), 1-50.
[6] O. Gonzalez and J. H. Maddocks, Global curvature, thickness, and the ideal shapes of knots, Proc. Natl. Acad. Sci. USA 96 (1999), no. 9, 4769-4773.
[7] Zheng-Xu He, On the Minimizers of the Mobuis Cross Energy of Links, AMS (2000).
[8] R. A. Litherland, J. Simon, O. Durumeric and E. Rawdon, Thickness of knots, Topology Appl. 91 (1999), 233-244.
[9] C. C. Lin and H. R. Schwetlick, On the geometric flow of Kirchhoff elastic rods, SIAM J. Appl. Math. 65 (2005), no. 2, 720-736.
[10] J. O’Hara, Energy of knots and conformal geometry, Series on Knots and Everything, 33. World Scientific Publishing Co., Inc., River Edge, NJ, 20003.
[11] J. O’Hara, Energy functionals of knots. II., Topology Appl. 56 (1994), no. 1, 45-61.
[12] J. M. Sullivan, Approximating ropelength by energy functions, 181–186, Contemp. Math., 304, (2002).