between the boundaries of the cylinder is mapped to the minimal length line between the tangent and minus the tangent on $S^{2}$, with appropriate sign) is the entire ball, plus the area $\beta$ covers with the two discs (and here ‘minimal’ means simply the minimal such positive area). In any event, modulo $4\pi$ the correction terms are equal. The isomorphism between the two correction terms means that in a very real sense choosing a swaddling map $\beta$ along with the homotopy class in which it sits is exactly the same thing as choosing a framing that is null-homotopic as an element of $SO(3)$. We have shown then that $\beta$ can be lifted to $\sigma$, but only for ‘half’ our possible choices of $\phi$.

So which “half” is it?
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We have shown then that for framings which give us an element of the trivial homotopy class of $SO(3)$, $\tau_{\phi}(\gamma)=\tau_{\beta}(\gamma)$. We have yet to show what framings those are.

Let us recall the mappings defined in section 2.3. $\overline{\Phi_{H}}$ mapped $S^{1}\times I$ to $S^{2}$, sending the two components of the boundary to the tangent bundle of the knot $\gamma$ in antipodal ways, while $\beta_{0}$ and $\beta_{1}$ took the boundary of a 