### - Art Gallery -

In mathematics (differential geometry) twist is the rate of rotation of a smooth ribbon around the space curve X=X(s), where s is the arc length of X and U=U(s) a unit 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 curve X' around and along the curve X. According to Love (1944) twist is defined by

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

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 $${\displaystyle T\in [0,1)}$$ and intrinsic twist $${\displaystyle N\in \mathbb {Z} }$$ 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) torsion becomes singular, but its singularity is integrable (Moffatt & Ricca 1992) and Tw remains continuous. This behavior has many important consequences for energy considerations in many fields of science.

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.

Twist (screw theory)
Twist (rational trigonometry)
twisted sheaf

References

Banchoff, T.F. & White, J.H. (1975) The behavior of the total twist and self-linking number of a closed space curve under inversions. Math. Scand. 36, 254–262.
Love, A.E.H. (1944) A Treatise on the Mathematical Theory of Elasticity. Dover, 4th Ed., New York.
Moffatt, H.K. & Ricca, R.L. (1992) Helicity and the Călugăreanu invariant. Proc. R. Soc. A 439, 411–429.