Linking Integral Projection

TL;DR

This paper demonstrates how the linking integral, an invariant of manifold links in Euclidean space, can be simplified to lower-dimensional integrals, broadening understanding of link invariants.

Contribution

It generalizes the reduction of the linking integral to lower dimensions for arbitrary manifolds, extending previous results beyond the classical Gauss integral.

Findings

Gauss integral in 3D reduces to a 2D winding number integral

Generalization to arbitrary manifolds under certain conditions

Simplification aids in computing linking invariants

Abstract

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.