Discrete thickness

Sebastian Scholtes

Molecular Based Mathematical Biology (2014)

  • Volume: 2, Issue: 1, page Article ID 1450045, 16 p.-Article ID 1450045, 16 p.
  • ISSN: 2299-3266

Abstract

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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.

How to cite

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Sebastian Scholtes. "Discrete thickness." Molecular Based Mathematical Biology 2.1 (2014): Article ID 1450045, 16 p.-Article ID 1450045, 16 p.. <http://eudml.org/doc/266712>.

@article{SebastianScholtes2014,
abstract = {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.},
author = {Sebastian Scholtes},
journal = {Molecular Based Mathematical Biology},
keywords = {discrete energy; thickness; ropelength; Γ-convergence; geometric knot theory; ideal knot; Schur’s Theorem; Möbius energy; polygonal knot; minimizers; knot energy; -convergence},
language = {eng},
number = {1},
pages = {Article ID 1450045, 16 p.-Article ID 1450045, 16 p.},
title = {Discrete thickness},
url = {http://eudml.org/doc/266712},
volume = {2},
year = {2014},
}

TY - JOUR
AU - Sebastian Scholtes
TI - Discrete thickness
JO - Molecular Based Mathematical Biology
PY - 2014
VL - 2
IS - 1
SP - Article ID 1450045, 16 p.
EP - Article ID 1450045, 16 p.
AB - 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.
LA - eng
KW - discrete energy; thickness; ropelength; Γ-convergence; geometric knot theory; ideal knot; Schur’s Theorem; Möbius energy; polygonal knot; minimizers; knot energy; -convergence
UR - http://eudml.org/doc/266712
ER -

References

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