In mathematics (differential geometry) twist is 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
′
=
X
+
ε
U
the twist (or total twist number)
T
w
measures the average winding of the curve
X
′
around and along the curve
X
. According to Love (1944) twist is defined by
T
w
=
1
2
π
∫
(
d
U
d
s
×
U
)
⋅
d
X
d
s
d
s
,
where
d
X
/
d
s
is the unit tangent vector to
X
. The total twist number
T
w
can be decomposed (Moffatt & Ricca 1992) into normalized total torsion
T
∈
[
0
,
1
)
and intrinsic twist
N
∈
Z
as
T
w
=
1
2
π
∫
τ
d
s
+
[
Θ
]
X
2
π
=
T
+
N
,
where
τ
=
τ
(
s
)
is the torsion of the space curve
X
, and
[
Θ
]
X
denotes the total rotation angle of
U
along
X
. Neither
N
nor
T
w
are independent of the ribbon field
U
. Instead, only the normalized torsion
T
is an invariant of the curve
X
(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
T
w
remains continuous. This behavior has many important consequences for energy considerations in many fields of science.
Together with the writhe
W
r
of
X
, twist is a geometric quantity that plays an important role in the application of the Călugăreanu–White–Fuller formula
L
k
=
W
r
+
T
w
in topological fluid dynamics (for its close relation to kinetic and magnetic helicity of a vector field), physical knot theory, and structural complexity analysis.
