storing framing information on the boundary, and between eliminating the boundary of $\overline{C_{2}(S^{1})}$ by the “$C$-swaddling” method as we have done in this section.

### Every framing gives a $C$-swaddling

The integral on the bottom splits into two parts— the “normal” pieces in which we have just the standard Gauss self-linking integral, and the “bumps”. On the bumps, $\theta$ goes from the tangent to the normal to minus the tangent— i.e. it traces out the image of a line between two points of the form $\{x\}\times
\{1\}$, $\{x\}\times \{0\}$ on $S^{1}\times I$, thus defining a $C$-swaddling, as the path can be assumed to be a great circle for all point on the knot.

We thus see that the “$C$-swaddling” construction gives us $\tau_{\phi}$, in just the same way as the construction in the first section gave us $\tau_{\beta}$.

### Every $C$-swaddling gives a framing

The fact that every $C$-swaddling gives a framing follows from the following lemma.

\[E:matz\] Every $C$-swaddling map is homotopic via $C$-swaddling maps to a map in which the path on $S^{2}$ from $\dot{\gamma}(s)$ to $-\dot{\gamma}(s)$ for each $s$ in the knot is a great circle between the two points.

1.  Every $C$-swaddling map is homotopic to 