Linking coefficient

From Wikipedia, the free encyclopedia

In mathematics, in the area of knot theory, the linking coefficient is a knot invariant that assigns an integer to a pair of closed curves. A non-zero value for this integer is sufficient to demonstrate that the curves are linked; however, non-trivial knots may have a zero linking coefficient. The linking coefficient was introduced by Gauss.

Contents

[edit] Definition

Given two closed curves \gamma_1:\mathbb{R}\to\mathbb{R}^3 and \gamma_2:\mathbb{R}\to\mathbb{R}^3, such that γ(t + 2π) = γ(t), the linking coefficient is defined as

\{\gamma_1,\gamma_2\}=\frac{1}{4\pi} \oint_{\gamma_1}\oint_{\gamma_2} \frac{[d\gamma_1(t)\times d\gamma_2(t)] \cdot (\gamma_1-\gamma_2)} {|\gamma_1(t)-\gamma_2(t)|^3}

Here, \times is the three-dimensional cross-product, while \cdot is the three-dimensional dot product.

[edit] Properties and examples

The linking coefficient is always an integer.

The Hopf link (two simple connected links) has a linking number of one.

Taking one curve to be the z-axis, while confining the other curve to the x-y plane, the linking coefficient is precisely the winding number; it is equal to the number of times the second curve winds around the z-axis.

[edit] Generalizations

The linking coefficient generalizes to higher dimensions, for any pair of cycles in n-dimensional space, when their dimension adds up to n − 1.

[edit] See also

[edit] References

Retrieved from "http://en.wikipedia.org../../../l/i/n/Linking_coefficient.html"