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