Clayton Shonkwiler

An elementary way of defining a linking integral

In our paper “Higher-dimensional linking integrals”, David Shea Vela-Vick and I found Gauss-type linking integrals in spheres of all dimensions.

Specifically, the Gauss linking integral computes the linking number between two parametrized closed curves in \(\mathbb{R}^3\), and easily generalizes to compute the linking number between a \(k\)-cycle and an \((n-k-1)\)-cycle in \(\mathbb{R}^n\).

However, finding Gauss-type linking integrals even in the three-dimensional sphere is much trickier. In our paper, we found such integrals on all spheres by considering the \(n\)-sphere embedded in \(\mathbb{R}^{n+1}\). By coning and capping, we tranform a \(k\)-cycle in the sphere into a \((k+1)\)-cycle in \(\mathbb{R}^{n+1}\) without changing the linking number. Then applying the Gauss integral in \(\mathbb{R}^{n+1}\), pushing the cap off to infinity, and integrating out the cone direction yields a linking integral which is intrinsic to the sphere.

The cone-and-cap process is shown above on the 2-sphere, and the situation in higher dimensions is analogous.