analogous to the linking number of a two component link. But in the case of a knot we run into the problem that $\Phi(x,x)$ is not defined (how do we define the direction from a point to itself?). So rather than $\Phi$ being a function from $S^{1}\times S^{1}$ to $S^{2}$ it is instead a function from $C_{2}(S^{1}):=\{(x,y)\in S^{1}\times S^{1}\mid x\neq y\}$ (the configuration space of two points on $S^{1}$) to $S^{2}$. The natural way to transport the Gauss integral to the case of a knot is then $$\eta(\gamma):=\frac{1}{4\pi}\int\limits_{C_{2}(S^{1})}\Phi^{*}
\omega$$ The problem now is that since $C_{2}(S^{1})$ is not a compact space, we are not guaranteed that the integral converges. There are two ways that we might try to solve this problem.

 

1.  We could compactify the configuration space, and examine by how much $\eta(\gamma)$ ‘fails’ to be invariant. By Stokes’ Theorem, we find that this quantity depends on the boundary of the compactified configuration space. We seek to eliminate this boundary by pasting some extra discs $D_{0}$ and $D_{1}$ onto our space, thus renormalizing the integral. Our invariant will then be $\eta(\gamma)$ plus a correction term which will depend on a “swaddling” map $\beta$, the extension of $\Phi$ to the 