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

Properties and implications

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

The ribbon concept plays an important role in the Călugăreanu-White-Fuller formula,^[3] that states that

[math]\displaystyle{ Lk = Wr + Tw , }[/math]

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

Bibliography

• Adams, Colin (2004), The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, American Mathematical Society, ISBN 0-8218-3678-1 
• Călugăreanu, Gheorghe (1959), "L'intégrale de Gauss et l'analyse des nœuds tridimensionnels", Revue de Mathématiques Pure et Appliquées 4: 5–20 
• 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 
• 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 

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