Twist (differential Geometry)

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 $U=U(s)$ a unit normal vector, perpendicular at each point to $(s)$. 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

:$Tw = \dfrac \int \left( U \times \dfrac \right) \cdot \dfrac ds \; ,$

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 torsion of the space curve $X$, and \left[ \Theta \right">Torsion of a curve">torsion of the space curve $X$, and $\left[ \Theta \rightX$ 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.

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.
*Goriely, A. (2006) Twisted elastic rings and the rediscoveries of Michell’s instability. ''J Elasticity'' 84, 281-299.
* 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 Calugareanu invariant. ''Proc. R. Soc. London A'' 439, 411-429. Also in: (1995) Knots and Applications (ed. L.H. Kauffman), pp. 251-269. World Scientific.
* Ricca, R.L. (1997) Evolution and inflexional instability of twisted magnetic flux tubes. ''Solar Physics'' 172, 241-248.
* Ricca, R.L. (2005) Inflexional disequilibrium of magnetic flux tubes. ''Fluid Dynamics Research'' 36, 319-332.

Differential geometry
Topology