Twist (mathematics)

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


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

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


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

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

When the ribbon is deformed so as to pass through an inflectional state (i.e. $$X$$ has a point of inflection), the torsion $$\tau$$ becomes singular. The total torsion $$T$$ jumps by $$\pm 1$$ and the total angle $$N$$ simultaneously makes an equal and opposite jump of $$ \mp 1 $$ (Moffatt & Ricca 1992) and $$Tw$$ 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 $$Wr$$ of $$X$$, twist is a geometric quantity that plays an important role in the application of the Călugăreanu–White–Fuller formula $$Lk = Wr + Tw$$ in topological fluid dynamics (for its close relation to kinetic and magnetic helicity of a vector field), physical knot theory, and structural complexity analysis.