we must paste on should consist of pairs $(x,\theta)$ in which we store the “address” $x$ of the point, as well as the direction from which $x$ is coming in on $\gamma_{\varepsilon}$. We may depict the newly created boundary of our compactified space as a continuum of pieces which can be schematically depicted like this:

$$\xy
0;/r3pc/: (2,0)="a"; 0 **@{-}; (0,.1)="b"; (.8,.1)="c" **@{-};
(1.2,.1)="d" **\crv{(1,.4)}; (2,.1)="f" **@{-}
\endxy$$

The leftmost point of the semicircle corresponds to a point $\gamma_{\varepsilon}(t)$ coming in to $\gamma_{t}$ on a backward sweeping tangent, the rightmost one corresponds to the point coming in on a forward sweeping tangent, and the apex corresponds to the point coming in straight off the normal. The tangent and the normal at each point define the framing for the knot, so that we see that we have not lost any information. Let us define $$\tau_{\phi}:=\frac{1}{2\pi}\int_{\mathbb{L}^{\overrightarrow{n}}(S^{1})_{0}}\overline{\Phi^{*}_{\gamma}}\omega$$

Our main claim in this section is:

$\eta+\tau_{\phi}$ is an invariant of framed knots.

To see that $\eta+\tau_{\phi}$ is in fact the limit of the integral of the pullback of the volume form, note that the domain at every $\varepsilon >0$ where $\varepsilon$ is fixed is cobordant to the integral at ‘the bottom’, where $\varepsilon=0$, and 