Writhing number

Let $ C $ be a closed imbedded smooth curve in $ \mathbf R ^ {3} $. For each ordered pair of points $ x, y $ on $ C $, let $ e ( x, y) = ( y- x) / \| y- x \| $ be the unit-length vector pointing from $ x $ to $ y $. This gives a mapping $ e: C \times C \rightarrow S ^ {2} $. The writhing number of the space curve $ C $ is

$$ \mathop{\rm Wr} ( C) = \frac{1}{4 \pi } \int\limits _ {C \times C } e ^ {*} d \Sigma , $$

where $ e ^ {*} d \Sigma $ is the pull-back along $ e $ of the standard area element $ d \Sigma $ on the unit sphere $ S ^ {2} $. In terms of local curve parameters $ s _ {1} $ and $ s _ {2} $ at $ x $ and $ y $ it can be described as

$$ \mathop{\rm Wr} ( C) = \frac{1}{4 \pi } \int\limits \int\limits \left ( \frac{\partial e }{\partial s _ {1} } \times \frac{\partial e }{\partial s _ {2} } \cdot e \right ) ds _ {1} ds _ {2} . $$

Now let $ R $ be a ribbon based on $ C $. Such a ribbon is obtained by taking a smooth vector field $ v $ of unit-length vectors on $ C $ such that $ v( x) $ is always perpendicular to the tangent vector along $ C $ at $ x \in C $. The unit length is chosen small enough such that each unit-length line segment $ v( x) $ at $ x $ only intersects $ C $ at $ x $. The ribbon $ R $ is the union of all the closed unit-length line segments $ v( x) $, $ x \in C $. Let $ C ^ \prime $ be the smooth curve of end points of the $ v( x) $. The total twist of the ribbon $ R $ is defined as

$$ \mathop{\rm Tw} ( R) = \frac{1}{2 \pi } \int\limits _ { C } v ^ \perp \cdot dv , $$

where $ v ^ \perp $ at $ x \in C $ is the unit-length vector such that $ v $, $ v ^ \perp $ and $ t $, the unit-length tangent vector along $ C $ at $ x $, make up a right-handed orthonormal $ 3 $- frame. The writhing number of $ C $, total twist of $ R $ and the linking number $ \mathop{\rm Lk} ( C, C ^ \prime ) $( cf. Linking coefficient), given by the Gauss formula

$$ \mathop{\rm Lk} ( C, C ^ \prime ) = \frac{1}{4 \pi } \int\limits _ {C \times C ^ \prime } e ^ {*} d \Sigma $$

(where now $ x $ runs over $ C $ and $ y $ over $ C ^ \prime $), are related by White's formula:

$$ \mathop{\rm Lk} ( C , C ^ \prime ) = \mathop{\rm Tw} ( R) + \mathop{\rm Wr} ( C) . $$

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