Grace–Walsh–Szegő theorem

In mathematics (differential geometry) twist denotes 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 $X^{\prime }=X+\epsilon U$ the twist (or total twist number) $Tw$ measures the average winding of the curve $X^{\prime }$ around and along the curve $X$. According to Love (1944) twist is defined by

$Tw=\frac{1}{2\pi }\int (\frac{dU}{ds}\times U)\cdot \frac{dX}{ds}ds,$

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 $T$ and intrinsic twist $N$, that is

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

where $\tau =\tau (s)$ is the torsion of the space curve $X$, and ${\left[\mathrm{\Theta }\right]}_X$ denotes the total rotation angle of $U$ along $X$. The total twist number $Tw$ depends on the choice of the vector field $U$ (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.

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.