User:Michael Hardy/Gauss link invariant

From a posting to the usenet newsgroup sci.math.research on November 17th, 1994 by Paulo Ney de Souza, approved by moderator Greg Kuperberg:

I am looking for a good reference (preferably a textbook) for the Gaussian integral invariance under isotopy. That is the fact that the integral


 * $$ {\ell}k(\kappa_1,\kappa_2)

= \frac{1}{4\pi} \int_{\kappa_1} \int_{\kappa_2} \frac{\langle dv_1 \times dv_2, v_1 - v_2 \rangle}{\left| v_1 - v_2 \right|^3} $$

is an integer and invariant unnder isotopy of the components of a link and of metric choices.

One would hope that this would be all over textbooks in Diff. Topology and some text in Knot Theory, but apparently it is NOT! The only one where I could find it stated [DFN] they wave hands to early and makes it difficult for an undergraduate to unnderstand.

If anyone knows a good referencec I would like to hear about it.


 * [BG] Berger, M. & Gostiaux, B., Differential geometry: manifolds, curves, and surfaces. Springer-Verlag, 1988


 * [DFN] Dubrovin, B. A., Fomenko, A. T., & Novikov S. P. Modern geometry--methods and applications, vol 2, Springer-Verlag, 1992

Paulo Ney de Souza desouza[at]math.berkeley.edu

[mod note: The proof that I know that is it invariant unnder isotopy is that, if you let F(v1) be the result of integrating over v2 in &kappa;2, then F(v1) is curl-free except that it is singular on &kappa;2. The line integral of F(v1) is therefore the same by Stokes' theorem on &kappa;1 and &kappa;1&prime; as long as there is an annulus connecting &kappa;1 and &kappa;1&prime; that does not cross &kappa;2. The annulus can even cross itself; the linking number between &kappa;1 and &kappa;2 does not change if you make &kappa;1 cross itself. Switching &kappa;1 and &kappa;2, it's also constant as you vary &kappa;2.

The proof that you get an integer is similarly asymmetric. Imagine an annulus that connects &kappa;1 and &kappa;1&prime; that does cross &kappa;2. The surface integral on this annulus, which is the difference between line integrals on &kappa;1 and &kappa;1&prime;, is the integral of a bunch of delta functions at the places where it crosses &kappa;2. The surface integral does not depend on the surface, of course; putting it in favorable position, namely perpendicular to &kappa;2, it is easy to see that these delta functions have integral integrals.

However, this is not a reference. Maybe the best place to look is an E&M physics book, since the theorem is just the same as Ampere's law - Greg.