Twist (mathematics)


In mathematics 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 measures the average winding of the curve around
and along the curve. According to Love twist is defined by
where is the unit tangent vector to.
The total twist number can be decomposed 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 .
When the ribbon is deformed so as to pass through an inflectional state torsion becomes singular, but its singularity is integrable 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, physical knot theory, and structural complexity analysis.