Linking number


In mathematics, the linking number is a numerical invariant that describes the linking of two closed curves in three-dimensional space. Intuitively, the linking number represents the number of times that each curve winds around the other. The linking number is always an integer, but may be positive or negative depending on the orientation of the two curves.
The linking number was introduced by Gauss in the form of the linking integral. It is an important object of study in knot theory, algebraic topology, and differential geometry, and has numerous applications in mathematics and science, including quantum mechanics, electromagnetism, and the study of DNA supercoiling.

Definition

Any two closed curves in space, if allowed to pass through themselves but not each other, can be moved into exactly one of the following standard positions. This determines the linking number:
Each curve may pass through itself during this motion, but the two curves must remain separated throughout. This is formalized as regular homotopy, which further requires that each curve be an immersion, not just any map. However, this added condition does not change the definition of linking number, which is an example of an h-principle, meaning that geometry reduces to topology.

Proof

This fact is most easily proven by placing one circle in standard position, and then showing that linking number is the only invariant of the other circle. In detail:
There is an algorithm to compute the linking number of two curves from a link diagram. Label each crossing as positive or negative, according to the following rule:

The total number of positive crossings minus the total number of negative crossings is equal to twice the linking number. That is:
where n1, n2, n3, n4 represent the number of crossings of each of the four types. The two sums and are always equal, which leads to the following alternative formula
The formula involves only the undercrossings of the blue curve by the red, while involves only the overcrossings.

Properties and examples

Given two non-intersecting differentiable curves, define the Gauss map from the torus to the sphere by
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 ' that goes to v under the Gauss map corresponds to a crossing in the link diagram where is over. Also, a neighborhood of ' 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 the Gauss map. 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 γ1 and γ2 enables an explicit formula as a double line integral, the Gauss linking integral:
This integral computes the total signed area of the image of the Gauss map and then divides by the area of the sphere.

In quantum field theory

In quantum field theory, Gauss' integral definition arises when computing the expectation value of the Wilson loop observable in Chern–Simons gauge theory. Explicitly, the abelian Chern–Simons action for a gauge potential one-form on a three-manifold is given by
We are interested in doing the Feynman path integral for Chern–Simons in :
Here, is the antisymmetric symbol. Since the theory is just Gaussian, no ultraviolet regularization or renormalization is needed. Therefore, the topological invariance of right hand side ensures that the result of the path integral will be a topological invariant. The only thing left to do is provide an overall normalization factor, and a natural choice will present itself. Since the theory is Gaussian and abelian, the path integral can be done simply by solving the theory classically and substituting for.
The classical equations of motion are
Here, we have coupled the Chern–Simons field to a source with a term in the Lagrangian. Obviously, by substituting the appropriate, we can get back the Wilson loops. Since we are in 3 dimensions, we can rewrite the equations of motion in a more familiar notation:
Taking the curl of both sides and choosing Lorenz gauge, the equations become
From electrostatics, the solution is
The path integral for arbitrary is now easily done by substituting this into the Chern–Simons action to get an effective action for the field. To get the path integral for the Wilson loops, we substitute for a source describing two particles moving in closed loops, i.e., with
Since the effective action is quadratic in, it is clear that there will be terms describing the self-interaction of the particles, and these are uninteresting since they would be there even in the presence of just one loop. Therefore, we normalize the path integral by a factor precisely cancelling these terms. Going through the algebra, we obtain
where
which is simply Gauss' linking integral. This is the simplest example of a topological quantum field theory, where the path integral computes topological invariants. This also served as a hint that the nonabelian variant of Chern–Simons theory computes other knot invariants, and it was shown explicitly by Edward Witten that the nonabelian theory gives the invariant known as the Jones polynomial.
The Chern-Simons gauge theory lives in 3 spacetime dimensions. More generally, there exists higher dimensional topological quantum field theories. There exists more complicated multi-loop/string-braiding statistics of 4-dimensional gauge theories captured by the link invariants of exotic topological quantum field theories in 4 spacetime dimensions.

Generalizations