Writhing number
Let be a closed imbedded smooth curve in
. For each ordered pair of points
on
, let
be the unit-length vector pointing from
to
. This gives a mapping
. The writhing number of the space curve
is
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where is the pull-back along
of the standard area element
on the unit sphere
. In terms of local curve parameters
and
at
and
it can be described as
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Now let be a ribbon based on
. Such a ribbon is obtained by taking a smooth vector field
of unit-length vectors on
such that
is always perpendicular to the tangent vector along
at
. The unit length is chosen small enough such that each unit-length line segment
at
only intersects
at
. The ribbon
is the union of all the closed unit-length line segments
,
. Let
be the smooth curve of end points of the
. The total twist of the ribbon
is defined as
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where at
is the unit-length vector such that
,
and
, the unit-length tangent vector along
at
, make up a right-handed orthonormal
-frame. The writhing number of
, total twist of
and the linking number
(cf. Linking coefficient), given by the Gauss formula
![]() |
(where now runs over
and
over
), are related by White's formula:
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This formula has applications to the coiling and supercoiling of DNA.
References
[a1] | W.F. Pohl, "DNA and differential geometry" Math. Intelligencer , 3 (1980) pp. 20–27 |
[a2] | J.H. White, "Self-linking and the Gauss integral in higher dimensions" Amer. J. Math. , 91 (1969) pp. 693–728 |
Writhing number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Writhing_number&oldid=15244