# Twist (mathematics)

In mathematics (differential geometry) **twist** denotes the rate of rotation of a smooth ribbon around the space curve , where is the arc-length of and a unit vector perpendicular at each point to . Since the ribbon has edges and the twist (or *total twist number*) measures the average winding of the curve around
and along the curve . According to Love (1944) twist is defined by

where is the unit tangent vector to .
The total twist number can be decomposed (Moffatt & Ricca 1992) into *normalized total torsion* and *intrinsic twist* , that is

where is the torsion of the space curve , and denotes the total rotation angle of along . The total twist number depends on the choice of the vector field (Banchoff & White 1975).

When the ribbon is deformed so as to pass through an *inflectional state* (i.e. has a point of inflection) torsion becomes singular, but its singularity is integrable (Moffatt & Ricca 1992) and remains continuous. This behavior has many important consequences for energy considerations in many fields of science.

Together with the writhe of , twist is a geometric quantity that plays an important role in the application of the Călugăreanu–White–Fuller formula 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.