Ribbon (mathematics)

In mathematics (differential geometry) by a ribbon (or strip) $(X,U)$ is meant a smooth space curve $X$ given by a three-dimensional vector $X(s)$ , depending continuously on the curve arc-length $s$ ( $a\leq s\leq b$ ), together with a smoothly varying unit vector $U(s)$ perpendicular to $X$ at each point (Blaschke 1950).

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

The ribbon concept plays an important role in the Cǎlugǎreǎnu-White-Fuller formula (Fuller 1971), that states that

Lk=Wr+Tw\;,

where $Lk$ is the asymptotic (Gauss) linking number (a topological quantity), $Wr$ denotes the total writhing number (or simply writhe) and $Tw$ is the total twist number (or simply twist).

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.

References

Blaschke, W. (1950) Einführung in die Differentialgeometrie. Springer-Verlag. ISBN 9783817115495
Fuller, F.B. (1971) The writhing number of a space curve. PNAS USA 68, 815-819.

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