Twist (differential geometry)

In differential geometry, the twist of a ribbon is its rate of axial rotation. Let a ribbon ${\displaystyle (X,U)}$ be composed of a space curve, ${\displaystyle X=X(s)}$, where ${\displaystyle s}$ is the arc length of ${\displaystyle X}$, and ${\displaystyle U=U(s)}$ the a unit normal vector, perpendicular at each point to ${\displaystyle X}$. Since the ribbon ${\displaystyle (X,U)}$ has edges ${\displaystyle X}$ and ${\displaystyle X'=X+\varepsilon U}$, the twist (or total twist number) ${\displaystyle Tw}$ measures the average winding of the edge curve ${\displaystyle X'}$ around and along the axial curve ${\displaystyle X}$. According to Love (1944) twist is defined by

${\displaystyle Tw={\dfrac {1}{2\pi }}\int \left(U\times {\dfrac {dU}{ds}}\right)\cdot {\dfrac {dX}{ds}}ds\;,}$

where ${\displaystyle dX/ds}$ is the unit tangent vector to ${\displaystyle X}$. The total twist number ${\displaystyle Tw}$ can be decomposed (Moffatt & Ricca 1992) into normalized total torsion ${\displaystyle T\in [0,1)}$ and intrinsic twist ${\displaystyle N\in \mathbb {Z} }$ as

${\displaystyle Tw={\dfrac {1}{2\pi }}\int \tau \;ds+{\dfrac {\left[\Theta \right]_{X}}{2\pi }}=T+N\;,}$

where ${\displaystyle \tau =\tau (s)}$ is the torsion of the space curve ${\displaystyle X}$, and ${\displaystyle \left[\Theta \right]_{X}}$ denotes the total rotation angle of ${\displaystyle U}$ along ${\displaystyle X}$. Neither ${\displaystyle N}$ nor ${\displaystyle Tw}$ are independent of the ribbon field ${\displaystyle U}$. Instead, only the normalized torsion ${\displaystyle T}$ is an invariant of the curve ${\displaystyle X}$ (Banchoff & White 1975).

When the ribbon is deformed so as to pass through an inflectional state (i.e. ${\displaystyle X}$ has a point of inflection), the torsion ${\displaystyle \tau }$ becomes singular. The total torsion ${\displaystyle T}$ jumps by ${\displaystyle \pm 1}$ and the total angle ${\displaystyle N}$ simultaneously makes an equal and opposite jump of ${\displaystyle \mp 1}$ (Moffatt & Ricca 1992) and ${\displaystyle Tw}$ remains continuous. This behavior has many important consequences for energy considerations in many fields of science (Ricca 1997, 2005; Goriely 2006).

Together with the writhe ${\displaystyle Wr}$ of ${\displaystyle X}$, twist is a geometric quantity that plays an important role in the application of the Călugăreanu–White–Fuller formula ${\displaystyle Lk=Wr+Tw}$ in topological fluid dynamics (for its close relation to kinetic and magnetic helicity of a vector field), physical knot theory, and structural complexity analysis.