# Twist (mathematics)

In mathematics (differential geometry) **twist** is 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*
as

where is the torsion of the space curve , and denotes the total rotation angle of along . Neither nor are independent of the ribbon field . Instead, only the normalized torsion is an invariant of the curve (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.