# Discrete thickness

Molecular Based Mathematical Biology (2014)

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

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topSebastian 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

top- [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. Zbl1036.57001
- [3] X. Dai and Y. Diao. The minimum of knot energy functions. J. Knot Theory Ramifications, 9(6):713–724, 2000. Zbl0999.57011
- [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. Zbl0089.38402
- [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. Zbl1028.57008
- [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. Zbl1057.57500
- [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. Zbl1006.49001
- [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. Zbl0924.57011
- [13] A. Lytchak. Almost convex subsets. Geom. Dedicata, 115:201–218, 2005. Zbl1088.53025
- [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] Zbl1152.57005
- [15] J. W. Milnor. On the total curvature of knots. Ann. of Math. (2), 52:248–257, 1950. Zbl0037.38904
- [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. Zbl0943.57002
- [18] E. J. Rawdon. Approximating smooth thickness. J. Knot Theory Ramifications, 9(1):113–145, 2000. Zbl0999.57010
- [19] E. J. Rawdon. Can computers discover ideal knots? Experiment. Math., 12(3):287–302, 2003. Zbl1073.57003
- [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. Zbl0864.92011
- [26] A. Stasiak, V. Katritch, and L. H. Kauffman. Ideal knots, volume 19 of Series on Knots and Everything. World Scientific Publishing Co. Inc., River Edge, NJ, 1998. Zbl0915.00018
- [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] Zbl1282.49034
- [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. Zbl1151.53305

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