Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
2014 | 2 | 1 |
Tytuł artykułu

Discrete thickness

Treść / Zawartość
Warianty tytułu
Języki publikacji
We investigate the relationship between a discrete version of thickness and its smooth counterpart. These discrete energies are deffned on equilateral polygons with n vertices. It will turn out that the smooth ropelength, which is the scale invariant quotient of length divided by thickness, is the Γ-limit of the discrete ropelength for n → ∞, regarding the topology induced by the Sobolev norm ‖ · ‖ W1,∞(S1,ℝd). This result directly implies the convergence of almost minimizers of the discrete energies in a fixed knot class to minimizers of the smooth energy.Moreover,we show that the unique absolute minimizer of inverse discrete thickness is the regular n-gon.
  • [1] J. Cantarella, J. H. Fu, R. Kusner, and J. M. Sullivan. Ropelength criticality. arxiv:1102.3234, 2011 (to appear in Geom.Topol.).
  • [2] J. Cantarella, R. B. Kusner, and J. M. Sullivan. On the minimum ropelength of knots and links. Invent. Math., 150(2):257–286, 2002.
  • [3] X. Dai and Y. Diao. The minimum of knot energy functions. J. Knot Theory Ramifications, 9(6):713–724, 2000.
  • [4] G. Dal Maso. An introduction to Γ-convergence. Progress in Nonlinear Differential Equations and their Applications, 8.Birkhäuser Boston Inc., Boston, MA, 1993.
  • [5] H. Federer. Curvature measures. Trans. Amer. Math. Soc., 93:418–491, 1959.
  • [6] H. Gerlach. Der Globale Krümmungsradius für offene und geschlossene Kurven im Rn. Diplomarbeit, Mathematisch-Naturwissenschaftliche Fakultät, Rheinische Friedrich-Wilhelms-Universität Bonn, 2004.
  • [7] O. Gonzalez and R. de la Llave. Existence of ideal knots. J. Knot Theory Ramifications, 12(1):123–133, 2003.
  • [8] O. Gonzalez and J. H. Maddocks. Global curvature, thickness, and the ideal shapes of knots. Proc. Natl. Acad. Sci. USA,96(9):4769–4773 (electronic), 1999.
  • [9] O. Gonzalez, J. H. Maddocks, F. Schuricht, and H. von der Mosel. Global curvature and self-contact of nonlinearly elasticcurves and rods. Calc. Var. Partial Differential Equations, 14(1):29–68, 2002.
  • [10] V. Katritch, J. Bednar, D. Michoud, R. G. Scharein, J. Dubochet, and A. Stasiak. Geometry and physics of knots. Nature,384(6605):142–145, 1996.
  • [11] V. Katritch, W. K. Olson, P. Pieranski, J. Dubochet, and A. Stasiak. Properties of ideal composite knots. Nature,388(6638):148–151, 1997.
  • [12] R. A. Litherland, J. Simon, O. Durumeric, and E. Rawdon. Thickness of knots. Topology Appl., 91(3):233–244, 1999.
  • [13] A. Lytchak. Almost convex subsets. Geom. Dedicata, 115:201–218, 2005.
  • [14] K. C. Millett, M. Piatek, and E. J. Rawdon. Polygonal knot space near ropelength-minimized knots. J. Knot Theory Ramifications,17(5):601–631, 2008.[WoS]
  • [15] J. W. Milnor. On the total curvature of knots. Ann. of Math. (2), 52:248–257, 1950.
  • [16] E. J. Rawdon. Thickness of polygonal knots. PhD thesis, University of Iowa, 1997.
  • [17] E. J. Rawdon. Approximating the thickness of a knot. In Ideal knots, volume 19 of Ser. Knots Everything, pages 143–150.World Sci. Publ., River Edge, NJ, 1998.
  • [18] E. J. Rawdon. Approximating smooth thickness. J. Knot Theory Ramifications, 9(1):113–145, 2000.
  • [19] E. J. Rawdon. Can computers discover ideal knots? Experiment. Math., 12(3):287–302, 2003.
  • [20] S. Scholtes. Discrete Möbius Energy. arxiv:1311.3056v3, 2013.
  • [21] S. Scholtes. On hypersurfaces of positive reach, alternating Steiner formulæ and Hadwiger’s Problem. arxiv:1304.4179,2013.
  • [22] F. Schuricht and H. von der Mosel. Global curvature for rectifiable loops. Math. Z., 243(1):37–77, 2003.
  • [23] F. Schuricht and H. von der Mosel. Characterization of ideal knots. Calc. Var. Partial Differential Equations, 19(3):281–305,2004.
  • [24] J. Simon. Physical knots. In Physical knots: knotting, linking, and folding geometric objects in R3 (Las Vegas, NV, 2001),volume 304 of Contemp. Math., pages 1–30. Amer. Math. Soc., Providence, RI, 2002.
  • [25] A. Stasiak, V. Katritch, J. Bednar, D. Michoud, and J. Dubochet. Electrophoretic mobility of DNA knots. Nature,384(6605):122, 1996.
  • [26] A. Stasiak, V. Katritch, and L. H. Kauffman. Ideal knots, volume 19 of Series on Knots and Everything. World ScientificPublishing Co. Inc., River Edge, NJ, 1998.
  • [27] P. Strzelecki, M. Szumanska, and H. von der Mosel. On some knot energies involving Menger curvature. Topology Appl.,160(13):1507–1529, 2013.[WoS]
  • [28] J. M. Sullivan. Approximating ropelength by energy functions. In Physical knots: knotting, linking, and folding geometricobjects in R3 (Las Vegas, NV, 2001), volume 304 of Contemp. Math., pages 181–186. Amer. Math. Soc., Providence, RI,2002.
  • [29] J. M. Sullivan. Curves of finite total curvature. In Discrete differential geometry, volume 38 of Oberwolfach Semin., pages137–161. Birkhäuser, Basel, 2008.
Typ dokumentu
Identyfikator YADDA
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.