Ribbon theory

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

Properties and implications

The ribbon $(X,U)$ is called ''simple'' if $X$ is a simple curve (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 Călugăreanu formula, that states that

:$Lk = Wr + Tw ,$

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

Bibliography

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* {{Citation , last=White , first=James H. , title=Self-linking and the Gauss integral in higher dimensions , journal=American Journal of Mathematics , volume=91 , issue=3 , pages=693–728 , year=1969 , doi=10.2307/2373348 , jstor=2373348 , mr=0253264

Differential geometry
Topology