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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.
See also

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


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.

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