# 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
How to Cite This Entry:
Writhing number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Writhing_number&oldid=49236