until we hit $p_{s}$ and continuing with $p_{s}$. $\varphi_{1}(p_{s})$ can be taken by homotopy to a curve in the image of the cylinder on $S^{2}$, which is $p_{s}$ with a Dehn twist added.

3.  Thus for each mapping $g$ considered in step 1, the map from the cylinder has a single representative $g$ composed with the identity. There are a $\mathbb{Z}$-worth of choices of $g$, corresponding to taking the path on the cylinder from $s\times {i}$, $i\in \mathbf{I}$, to the great circle which wraps around the circle $n$ times for all $n\in \mathbb{Z}$.

Choosing the midpoint for each such great circle gives us a normal to the knot at $s$. In other words, given a family of forward-sweeping tangents to the knot, a $C$-swaddling map gives us a smooth family of normals to the knot, thus giving the knot a framing.

Holonomy through $C$-swaddling
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To show that the correction term $\tau_{\phi}$ that we get is the total holonomy, we must first represent $\phi^{*}\omega$ as the pullback of an element of $SO(3)$. For this purpose, as the $\phi$ swaddling map is a smooth extension of the Gauss map, let us redefine $\overline{\Phi_{\gamma}}(x,y)$ to be $\Phi_{\gamma}(x,y)$ when $x\neq y$ and $\phi$ 