Ribbon theory

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Ribbon theory is a strand of mathematics within topology that deals with ribbons and has seen particular application as regards DNA.[1]

Ribbon[]

This section is an excerpt from Ribbon (mathematics).[]
In differential geometry, a ribbon (or strip) is the combination of a smooth space curve and its corresponding normal vector. More formally, a ribbon denoted by includes a curve given by a three-dimensional vector , depending continuously on the curve arc-length (), and a unit vector perpendicular to at each point.[2]

Concepts[]

  • Link (Lk) is the integer number of turns of the ribbon around its axis;
  • Twist (Tw) is the rate of rotation of the ribbon around its axis;
  • Writhe (Wr) is a measure of non-planarity of the ribbon's axis curve.

Work by Gheorghe Călugăreanu, James H. White, and F. Brock Fuller led to the Călugăreanu–White–Fuller theorem that .[3][4]

See also[]

References[]

  • Adams, Colin (2004), The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, American Mathematical Society, ISBN 0-8218-3678-1, MR 2079925
  • Călugăreanu, Gheorghe (1959), "L'intégrale de Gauss et l'analyse des nœuds tridimensionnels", Revue de Mathématiques Pure et Appliquées, 4: 5–20, MR 0131846
  • Călugăreanu, Gheorghe (1961), "Sur les classes d'isotopie des noeuds tridimensionels et leurs invariants", Czechoslovak Mathematical Journal, 11: 588–625, doi:10.21136/CMJ.1961.100486, MR 0149378
  • Fuller, F. Brock (1971), "The writhing number of a space curve", Proceedings of the National Academy of Sciences of the United States of America, 68 (4): 815–819, doi:10.1073/pnas.68.4.815, MR 0278197, PMC 389050, PMID 5279522
  • White, James H. (1969), "Self-linking and the Gauss integral in higher dimensions", American Journal of Mathematics, 91 (3): 693–728, doi:10.2307/2373348, JSTOR 2373348, MR 0253264

Notes[]

  1. ^ Vologodskiǐ, Aleksandr Vadimovich (1992). Topology and Physics of Circular DNA (First ed.). Boca Raton, FL. p. 49. ISBN 978-1138105058. OCLC 1014356603.
  2. ^ Blaschke, W. (1950) Einführung in die Differentialgeometrie. Springer-Verlag. ISBN 9783817115495
  3. ^ Dennis, Mark R.; Hannay, J.H (2005). "Geometry of Călugăreanu's theorem". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 461 (2062): 3245–3254. doi:10.1098/rspa.2005.1527. MR 2172227. S2CID 17766229.
  4. ^ Dennis, Mark. "The geometry of twisted ribbons". University of Bristol. Archived from the original on 3 May 2009. Retrieved 18 July 2010.


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