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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098170/w0981701.png" /> be a closed imbedded smooth curve in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098170/w0981702.png" />. For each ordered pair of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098170/w0981703.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098170/w0981704.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098170/w0981705.png" /> be the unit-length vector pointing from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098170/w0981706.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098170/w0981707.png" />. This gives a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098170/w0981708.png" />. The writhing number of the space curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098170/w0981709.png" /> is
+
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$#A+1 = 54 n = 0
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$#C+1 = 54 : ~/encyclopedia/old_files/data/W098/W.0908170 Writhing number
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098170/w09817010.png" /></td> </tr></table>
+
{{TEX|auto}}
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{{TEX|done}}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098170/w09817011.png" /> is the pull-back along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098170/w09817012.png" /> of the standard area element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098170/w09817013.png" /> on the unit sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098170/w09817014.png" />. In terms of local curve parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098170/w09817015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098170/w09817016.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098170/w09817017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098170/w09817018.png" /> it can be described as
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098170/w09817019.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm Wr} ( C)  =
 +
\frac{1}{4 \pi }
 +
\int\limits _ {C \times C } e  ^ {*}  d \Sigma ,
 +
$$
  
Now let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098170/w09817020.png" /> be a ribbon based on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098170/w09817021.png" />. Such a ribbon is obtained by taking a smooth vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098170/w09817022.png" /> of unit-length vectors on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098170/w09817023.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098170/w09817024.png" /> is always perpendicular to the tangent vector along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098170/w09817025.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098170/w09817026.png" />. The unit length is chosen small enough such that each unit-length line segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098170/w09817027.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098170/w09817028.png" /> only intersects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098170/w09817029.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098170/w09817030.png" />. The ribbon <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098170/w09817031.png" /> is the union of all the closed unit-length line segments <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098170/w09817032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098170/w09817033.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098170/w09817034.png" /> be the smooth curve of end points of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098170/w09817035.png" />. The total twist of the ribbon <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098170/w09817036.png" /> is defined as
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098170/w09817037.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm Wr} ( C)  =
 +
\frac{1}{4 \pi }
 +
\int\limits \int\limits \left (
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098170/w09817038.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098170/w09817039.png" /> is the unit-length vector such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098170/w09817040.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098170/w09817041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098170/w09817042.png" />, the unit-length tangent vector along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098170/w09817043.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098170/w09817044.png" />, make up a right-handed orthonormal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098170/w09817045.png" />-frame. The writhing number of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098170/w09817046.png" />, total twist of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098170/w09817047.png" /> and the linking number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098170/w09817048.png" /> (cf. [[Linking coefficient|Linking coefficient]]), given by the Gauss formula
+
\frac{\partial  e }{\partial  s _ {1} }
 +
\times
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098170/w09817049.png" /></td> </tr></table>
+
\frac{\partial  e }{\partial  s _ {2} }
 +
\cdot e \right )  ds _ {1}  ds _ {2} .
 +
$$
  
(where now <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098170/w09817050.png" /> runs over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098170/w09817051.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098170/w09817052.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098170/w09817053.png" />), are related by White's formula:
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098170/w09817054.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm Tw} ( R)  =
 +
\frac{1}{2 \pi }
 +
\int\limits _ { C } v  ^  \perp  \cdot dv ,
 +
$$
 +
 
 +
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
How to Cite This Entry:
Writhing number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Writhing_number&oldid=49236