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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 links may have a zero linking coefficient. The linking coefficient was introduced by Gauss in the form of a linking integral.
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Given two disjoint closed curves 
and 
, such that γ(t + 2π) = γ(t), the linking coefficient is defined by the Gauss linking integral
![\{\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}](../../../math/b/e/2/be2f19c23839e02da09bc5f26dbfd251.png)
Here, 
is the three-dimensional cross-product, while 
is the three-dimensional dot product.
Given two disjoint circles C1,C2 in 
, define a map from the torus to the sphere by
The degree of this map is the linking coefficient, and this map is called the Gauss map of the link.
This map can be constructed more abstractly thus: disjoint circles give a map from the product to configuration space:
The configuration space is homeomorphic to 
, which is homotopy equivalent to S2.
The analytic description above can be interpreted as: construct the map from the torus to the sphere as described here (this accounts for the (γ1 − γ2) / ( | γ1 − γ2 | )), then pull back the volume form on the sphere to the torus (the 1 / 4π is the volume form and the 
and the / ( | γ1 − γ2 | )2 are the Jacobian of the map) and integrate (these are the integrals) to find the degree.
This topological viewpoint can be used to extend the concept of linking coefficient to higher dimensions by considering pairs of cycles in n-dimensional space, when their dimension adds up to n − 1.
The linking coefficient is always an integer and is in fact the linking number.
The Hopf link (two simple connected links) has a linking number of one. The Whitehead link has zero linking number.
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.
- A.V. Chernavskii, "Linking coefficient" SpringerLink Encyclopaedia of Mathematics (2001)
- -, "Writhing number" SpringerLink Encyclopaedia of Mathematics (2001)