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 ${\displaystyle (X,U)}$ includes a curve ${\displaystyle X}$ given by a three-dimensional vector ${\displaystyle X(s)}$, depending continuously on the curve arc-length ${\displaystyle s}$ (${\displaystyle a\leq s\leq b}$), and a unit vector ${\displaystyle U(s)}$ perpendicular to ${\displaystyle {\partial X \over \partial s}(s)}$ at each point.^[1] Ribbons have seen particular application as regards DNA.^[2]

Properties and implications

The ribbon ${\displaystyle (X,U)}$ is called simple if ${\displaystyle X}$ is a simple curve (i.e. without self-intersections) and closed and if ${\displaystyle U}$ and all its derivatives agree at ${\displaystyle a}$ and ${\displaystyle b}$. For any simple closed ribbon the curves ${\displaystyle X+\varepsilon U}$ given parametrically by ${\displaystyle X(s)+\varepsilon U(s)}$ are, for all sufficiently small positive ${\displaystyle \varepsilon }$, simple closed curves disjoint from ${\displaystyle X}$.

The ribbon concept plays an important role in the Călugăreanu formula,^{[3] [4]}that states that

${\displaystyle Lk=Wr+Tw,}$

where ${\displaystyle Lk}$ is the asymptotic (Gauss) linking number, the integer number of turns of the ribbon around its axis; ${\displaystyle Wr}$ denotes the total writhing number (or simply writhe), a measure of non-planarity of the ribbon's axis curve; and ${\displaystyle Tw}$ is the total twist number (or simply twist), the rate of rotation of the ribbon around its axis.

Ribbon theory investigates geometric and topological aspects of a mathematical reference ribbon associated with physical and biological properties, such as those arising in topological fluid dynamics, DNA modeling and in material science.

See also

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

References

1. Blaschke, W. (1950) Einführung in die Differentialgeometrie. Springer-Verlag. ISBN 9783817115495
2. Vologodskiǐ, Aleksandr Vadimovich (1992). Topology and Physics of Circular DNA (First ed.). Boca Raton, FL. p. 49. ISBN 978-1138105058. OCLC 1014356603.
3. Fuller, F. Brock (1971). "The writhing number of a space curve" (PDF). Proceedings of the National Academy of Sciences of the United States of America. 68 (4): 815–819. Bibcode:1971PNAS...68..815B. doi:10.1073/pnas.68.4.815. MR 0278197. PMC 389050. PMID 5279522.
4. Dennis, M. R.; Hannay, J. H. (26 August 2005). "Geometry of Călugăreanu's theorem". Proceedings of the Royal Society A. 461 (2062): 3245–3254.

Bibliography

• 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
• 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