Editing Linking number (section)

==Gauss's integral definition==
Given two non-intersecting differentiable curves <math>\gamma_1, \gamma_2 \colon S^1 \rightarrow \mathbb{R}^3</math>, define the '''[[Carl Friedrich Gauss|Gauss]] map''' <math>\Gamma</math> from the [[torus]] to the [[unit sphere|sphere]] by
:<math>\Gamma(s,t) = \frac{\gamma_1(s) - \gamma_2(t)}{|\gamma_1(s) - \gamma_2(t)|}</math>

Pick a point in the unit sphere, ''v'', so that orthogonal projection of the link to the plane perpendicular to ''v'' gives a link diagram.  Observe that a point (''s'', ''t'') that goes to ''v'' under the Gauss map corresponds to a crossing in the link diagram where <math>\gamma_1</math> is over <math>\gamma_2</math>.  Also, a neighborhood of (''s'', ''t'') is mapped under the Gauss map to a neighborhood of ''v'' preserving or reversing orientation depending on the sign of the crossing.  Thus in order to compute the linking number of the diagram corresponding to ''v'' it suffices to count the ''signed'' number of times the Gauss map covers ''v''.  Since ''v'' is a [[regular value]], this is precisely the [[degree of a continuous mapping|degree]] of the Gauss map (i.e. the signed number of times that the [[image (mathematics)|image]] of Γ covers the sphere).  Isotopy invariance of the linking number is automatically obtained as the degree is invariant under homotopic maps.  Any other regular value would give the same number, so the linking number doesn't depend on any particular link diagram.

This formulation of the linking number of ''γ''<sub>1</sub> and ''γ''<sub>2</sub>  enables an explicit formula as a double [[line integral]], the '''Gauss linking integral''':

:<math>\begin{align}
  \operatorname{link}(\gamma_1,\gamma_2)
    &= \frac{1}{4\pi} \oint_{\gamma_1} \oint_{\gamma_2} \frac{\mathbf{r}_1 - \mathbf{r}_2}{|\mathbf{r}_1 - \mathbf{r}_2|^3} \cdot (d\mathbf{r}_1 \times d\mathbf{r}_2) \\[4pt]
    &= \frac{1}{4\pi} \int_{S^1 \times S^1} \frac{\det\left(\dot\gamma_1(s), \dot\gamma_2(t), \gamma_1(s) - \gamma_2(t)\right)}{\left|\gamma_1(s) - \gamma_2(t)\right|^3}\, ds\, dt
\end{align}</math>

This integral computes the total signed area of the image of the Gauss map (the integrand being the [[Jacobian matrix and determinant|Jacobian]] of Γ) and then divides by the area of the sphere (which is 4{{pi}}).