Linking Integral Projection

The linking integral is an invariant of the link-type of two manifolds immersed in a Euclidean space. It is shown that the ordinary Gauss integral in three dimensions may be simplified to a winding number integral in two dimensions. This result is then generalized to show that in certain circumstances the linking integral between arbitrary manifolds may be similarly reduced to a lower dimensional integral. 1. Reduction of the Gauss Integral to the Winding Number Integral The linking number of two disjoint oriented closed curves in R is an integer invariant that in some sense measures the extent of linking between the curves. While there are many equivalent ways to compute this number[3], the most wellknown is the linking integral of Gauss. In this section we show that this integral in 3-space may always be simplified to an integral in 2-space which is equivalent to a winding number integral. Proposition 1. Given two disjoint immersed closed curves s 7→ γ1(s) and t 7→ γ2(t) in R , the Gauss linking integral of the pair reduces to a sum of winding numbers of one curve about a sequence of points determined by the other, contained in some 2-dimensional hyperplane. Proof. The link of γ1 and γ2, lk(γ1, γ2), is given by the Gauss integral, lk(γ1, γ2) = 1 4π ∫