Difference between revisions of "Writhing number"
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+ | $#A+1 = 54 n = 0 | ||
+ | $#C+1 = 54 : ~/encyclopedia/old_files/data/W098/W.0908170 Writhing number | ||
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− | + | 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|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. | This formula has applications to the coiling and supercoiling of DNA. |
Latest revision as of 08:29, 6 June 2020
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 |
Writhing number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Writhing_number&oldid=49236