Writhing number

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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

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

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

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:

This formula has applications to the coiling and supercoiling of DNA.


[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
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