Twist (differential geometry)

For twists of curves in algebraic geometry, see twists of elliptic curves.

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.

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.