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Tytuł artykułu

Modeling repulsive forces on fibres via knot energies

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Modeling of repulsive forces is essential to the understanding of certain bio-physical processes, especially for the motion of DNA molecules. These kinds of phenomena seem to be driven by some sort of “energy” which especially prevents the molecules from strongly bending and forming self-intersections. Inspired by a physical toy model, numerous functionals have been defined during the past twenty-five years that aim at modeling self-avoidance. The general idea is to produce “detangled” curves having particularly large distances between distant strands. In this survey we present several families of these so-called knot energies. It turns out that they are quite similar from an analytical viewpoint. We focus on proving self-avoidance and existence of minimizers in every knot class. For a suitable subfamily of these energies we show how to prove that these minimizers are even infinitely differentiable
Słowa kluczowe
Wydawca
Rocznik
Tom
2
Numer
1
Opis fizyczny
Daty
otrzymano
2013-07-31
zaakceptowano
2014-04-09
online
2014-09-19
Twórcy
autor
  • Workgroup Applied Analysis, Karlsruhe Institute of Technology, Kaiserstraße 89-93, 76133 Karlsruhe, Germany,, simon.blatt@kit.edu
  • Fakultät für Mathematik der Universität Duisburg-Essen, Forsthausweg 2, 47057 Duisburg, Germany, philipp.reiter@uni-due.de
Bibliografia
  • [1] A. Abrams, J. Cantarella, J. H. G. Fu, M. Ghomi, and R. Howard. Circles minimize most knot energies. Topology, 42(2):381-394, 2003.[Crossref]
  • [2] S. Blatt. Note on continuously differentiable isotopies. Report 34, Institute for Mathematics, RWTH Aachen, August 2009.
  • [3] S. Blatt. The energy spaces of the tangent point energies. Preprint. To appear in Journal of Topology and Analysis, 2011.
  • [4] S. Blatt. Boundedness and regularizing effects of O’Hara’s knot energies. J. Knot Theory Ramifications, 21(1):1250010, 9, 2012.[WoS]
  • [5] S. Blatt. A note on integral Menger curvature for curves. Math. Nachr., 286(2-3):149-159, 2013.[WoS]
  • [6] S. Blatt and Ph. Reiter. Regularity theory for tangent-point energies: The non-degenerate sub-critical case. ArXiv e-prints, Aug. 2012.
  • [7] S. Blatt and Ph. Reiter. Stationary points of O’Hara’s knot energies. Manuscripta Mathematica, 140:29-50, 2013.[WoS]
  • [8] S. Blatt and Ph. Reiter. Towards a regularity theory of integral Menger curvature. In preparation, 2013.
  • [9] H. Brezis. How to recognize constant functions. A connection with Sobolev spaces. Uspekhi Mat. Nauk, 57(4(346)):59-74, 2002.
  • [10] G. Burde and H. Zieschang. Knots, volume 5 of de Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin, second edition, 2003.
  • [11] J. Cantarella, J. H. G. Fu, R. Kusner, and J. M. Sullivan. Ropelength Criticality. ArXiv e-prints, Feb. 2011.
  • [12] J. Cantarella, R. B. Kusner, and J. M. Sullivan. Tight knot values deviate from linear relations. Nature, 392:237-238, 1998.
  • [13] J. Cantarella, R. B. Kusner, and J. M. Sullivan. On the minimum ropelength of knots and links. Invent. Math., 150(2):257-286, 2002.
  • [14] M. H. Freedman, Z.-X. He, and Z. Wang. Möbius energy of knots and unknots. Ann. of Math. (2), 139(1):1-50, 1994.[WoS]
  • [15] S. Fukuhara. Energy of a knot. In A fête of topology, pages 443-451. Academic Press, Boston, MA, 1988.
  • [16] O. Gonzalez and R. de la Llave. Existence of ideal knots. J. Knot Theory Ramifications, 12(1):123-133, 2003.
  • [17] 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.[Crossref]
  • [18] 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.
  • [19] R. B. Kusner and J. M. Sullivan. Möbius-invariant knot energies. In Ideal knots, volume 19 of Ser. Knots Everything, pages 315-352. World Sci. Publishing, River Edge, NJ, 1998.
  • [20] H. K. Mofiatt. Pulling the knot tight. Nature, 384:114, 1996.
  • [21] J. O’Hara. Energy of a knot. Topology, 30(2):241-247, 1991.
  • [22] J. O’Hara. Family of energy functionals of knots. Topology Appl., 48(2):147-161, 1992.[Crossref]
  • [23] J. O’Hara. Energy functionals of knots. II. Topology Appl., 56(1):45-61, 1994.[Crossref]
  • [24] J. O’Hara. Energy of knots and conformal geometry, volume 33 of Series on Knots and Everything. World Scientific Publishing Co. Inc., River Edge, NJ, 2003.
  • [25] Ph. Reiter. All curves in a C1-neighbourhood of a given embedded curve are isotopic. Report 4, Institute for Mathematics, RWTH Aachen, October 2005.
  • [26] P. Strzelecki, M. Szumanska, and H. von der Mosel. Regularizing and self-avoidance effects of integral Menger curvature. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), IX(1):145-187, 2010.
  • [27] P. Strzelecki and H. von der Mosel. Tangent-point self-avoidance energies for curves. ArXiv e-prints, June 2010. Published in Journal of Knot Theory and Its Ramifications 21(05):1250044, 2012.
  • [28] D. W. Sumners. DNA, knots and tangles. In The mathematics of knots, volume 1 of Contrib. Math. Comput. Sci., pages 327-353. Springer, Heidelberg, 2011.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_mlbmb-2014-0004
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