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 ) {\displaystyle (X,U)} includes a curve X {\displaystyle X} given by a three-dimensional vector X ( s ) {\displaystyle X(s)} , depending continuously on the curve arc-length s {\displaystyle s} ( a ≤ s ≤ b {\displaystyle a\leq s\leq b} ), and a unit vector U ( s ) {\displaystyle U(s)} perpendicular to ∂ X ∂ s ( s ) {\displaystyle {\partial X \over \partial s}(s)} at each point. Ribbons have seen particular application as regards DNA.
Properties and implications The ribbon ( X , U ) {\displaystyle (X,U)} is called simple if X {\displaystyle X} is a simple curve (i.e. without self-intersections) and closed and if U {\displaystyle U} and all its derivatives agree at a {\displaystyle a} and b {\displaystyle b} . For any simple closed ribbon the curves X + ε U {\displaystyle X+\varepsilon U} given parametrically by X ( s ) + ε U ( s ) {\displaystyle X(s)+\varepsilon U(s)} are, for all sufficiently small positive ε {\displaystyle \varepsilon } , simple closed curves disjoint from X {\displaystyle X} . The ribbon concept plays an important role in the Călugăreanu formula, that states that
L k = W r + T w , {\displaystyle Lk=Wr+Tw,}
where L k {\displaystyle Lk} is the asymptotic (Gauss) linking number, the integer number of turns of the ribbon around its axis; W r {\displaystyle Wr} denotes the total writhing number (or simply writhe), a measure of non-planarity of the ribbon's axis curve; and T w {\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
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, hdl:10338.dmlcz/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
