Twist (mathematics)

In differential geometry, the twist of a ribbon is its rate of axial rotation. Let a ribbon [math]\displaystyle{ (X,U) }[/math] be composed of space curve [math]\displaystyle{ X=X(s) }[/math], where [math]\displaystyle{ s }[/math] is the arc length of [math]\displaystyle{ X }[/math], and [math]\displaystyle{ U=U(s) }[/math] the a unit normal vector, perpendicular at each point to [math]\displaystyle{ X }[/math]. Since the ribbon [math]\displaystyle{ (X,U) }[/math] has edges [math]\displaystyle{ X }[/math] and [math]\displaystyle{ X'=X+\varepsilon U }[/math], the twist (or total twist number) [math]\displaystyle{ Tw }[/math] measures the average winding of the edge curve [math]\displaystyle{ X' }[/math] around and along the axial curve [math]\displaystyle{ X }[/math]. According to Love (1944) twist is defined by

[math]\displaystyle{ Tw = \dfrac{1}{2\pi} \int \left( U \times \dfrac{dU}{ds} \right) \cdot \dfrac{dX}{ds} ds \; , }[/math]

where [math]\displaystyle{ dX/ds }[/math] is the unit tangent vector to [math]\displaystyle{ X }[/math]. The total twist number [math]\displaystyle{ Tw }[/math] can be decomposed (Moffatt & Ricca 1992) into normalized total torsion [math]\displaystyle{ T \in [0,1) }[/math] and intrinsic twist [math]\displaystyle{ N \in \mathbb{Z} }[/math] as

[math]\displaystyle{ Tw = \dfrac{1}{2\pi} \int \tau \; ds + \dfrac{\left[ \Theta \right]_X}{2\pi} = T+N \; , }[/math]

where [math]\displaystyle{ \tau=\tau(s) }[/math] is the torsion of the space curve [math]\displaystyle{ X }[/math], and [math]\displaystyle{ \left[ \Theta \right]_X }[/math] denotes the total rotation angle of [math]\displaystyle{ U }[/math] along [math]\displaystyle{ X }[/math]. Neither [math]\displaystyle{ N }[/math] nor [math]\displaystyle{ Tw }[/math] are independent of the ribbon field [math]\displaystyle{ U }[/math]. Instead, only the normalized torsion [math]\displaystyle{ T }[/math] is an invariant of the curve [math]\displaystyle{ X }[/math] (Banchoff & White 1975).

When the ribbon is deformed so as to pass through an inflectional state (i.e. [math]\displaystyle{ X }[/math] has a point of inflection), the torsion [math]\displaystyle{ \tau }[/math] becomes singular. The total torsion [math]\displaystyle{ T }[/math] jumps by [math]\displaystyle{ \pm 1 }[/math] and the total angle [math]\displaystyle{ N }[/math] simultaneously makes an equal and opposite jump of [math]\displaystyle{ \mp 1 }[/math] (Moffatt & Ricca 1992) and [math]\displaystyle{ Tw }[/math] 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 [math]\displaystyle{ Wr }[/math] of [math]\displaystyle{ X }[/math], twist is a geometric quantity that plays an important role in the application of the Călugăreanu–White–Fuller formula [math]\displaystyle{ Lk = Wr + Tw }[/math] 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

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.

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