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Tytuł artykułu

Discrete thickness

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Abstrakty

EN
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.

Twórcy

  • Institut für Mathematik, RWTH Aachen University, Templergraben 55, D-52062, Aachen, Germany

Bibliografia

  • [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.
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  • [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 elastic curves and rods. Calc. Var. Partial Differential Equations, 14(1):29–68, 2002.
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  • [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]
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  • [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.
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  • [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 geometric objects 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., pages 137–161. Birkhäuser, Basel, 2008.

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Bibliografia

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bwmeta1.element.doi-10_2478_mlbmb-2014-0005
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