Ribbon theory

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

Concepts

• Link is the integer number of turns of the ribbon around its axis;
• Twist is the rate of rotation of the ribbon around its axis;
• Writhe 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 Link = Writhe + Twist.[2][3]

See also

• Bollobás–Riordan polynomial
• Knots and graphs
• Knot theory
• DNA supercoil
• Möbius strip

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, 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: 815–819, doi:10.1073/pnas.68.4.815, MR 0278197, PMC 389050
• White, James H. (1969), "Self-linking and the Gauss integral in higher dimensions", American Journal of Mathematics, 91: 693–728, doi:10.2307/2373348, 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. 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.
3. Dennis, Mark. "The geometry of twisted ribbons". University of Bristol. Archived from the original on 3 May 2009. Retrieved 18 July 2010.