has to, since $\pi_{2}(S^{2})=\mathbb{Z}$, and so $\int_{D^{2}}\beta^{*}\omega$ assumes values in $4\pi\mathbb{Z}$).

Thus we find that $sl(\gamma):=\eta(\gamma)-\tau_{\beta}(\gamma)$ (the “self-linking number” of $\gamma$) is an integer-valued invariant of closed space curves along with a choice of homotopy class of swaddling maps $\beta$. But just how much information are we adding about the knot when we specify a homotopy class for a swaddling map?

Relating $\tau_{\beta}$ to the total torsion of a space curve
-------------------------------------------------------------

In passing, we may note that for a curve with nowhere vanishing curvature, $\tau_{\beta}$ corresponds to the notion of the correction term for the self-linking number as it was first defined by Calugareanu and later by Pohl, as the total torsion of a space curve. In section 2.4, we discovered that the correction term $\tau_{\beta}$ is equal, modulo $\mathbb{Z}$, to the area on $S_{2}$ covered by the map $\beta$. By the Gauss-Bonnet Theorem, this is equal to the total curvature of $\dot{\gamma}$. But if $\gamma$ has nowhere vanishing curvature, then this equals simply the total torsion of the space curve $\gamma$.

The holonomy construction
=========================

In the previous section, we transported the concept of a linking number from links to knots by compactifying the configuration space and pasting 