via this map, when restricted to $C$, will then by Stokes’ Theorem be equivalent to pulling back $de_{3}\wedge e_{2}$ via $\phi$ along $C$’s boundary. But here $\phi$ gives us the tangent, $e_{2}$ the normal, and $e_{3}$ the binormal, $\phi^{*}de_{3}\wedge e_{2}$ is the triple product $(\dot{\gamma},n,\dot{n})$. Here though we have $\int_{S^{1}}(\dot{\gamma},n,\dot{n})ds=
\int_{S^{1}}(\dot{\gamma},n,(n(s)+\dot{n}(s)ds))
=\int_{S^{1}}(\dot{\gamma},n,\dot{n}(s+ds))$. The last integral is measuring “by how far” the normal has strayed from its initial position at $t=0$ by the time we get to $t=L$. In other words, $\tau_{\phi}$ is measuring the **total holonomy** of the curve $\gamma$, with respect to the Reimannian connection on the normal bundle to the curve.

Equivalence to total torsion (again)
------------------------------------

Here again we have an easy proof that the correction term of the self-linking integral equals the total torsion. When our curve has a Frenet frame $(t,n,b)$ with curvature $\kappa$ and torsion $\tau$, the Frenet equations give us $$\tau_{\phi}=\frac{1}{2\pi}\int_{S^{1}}(\dot{\gamma},n,\dot{n})ds=
\frac{1}{2\pi}\int_{S^{1}}(\dot{\gamma},n,(\tau b-\kappa
t))ds=\frac{1}{2\pi}\int_{S^{1}}\tau ds \label{tort}$$

Equivalence of the two constructions
====================================

In the previous sections, we have presented two alternative ways of introducing a correction term to the Gauss self-linking integral for a knot, making it an invariant. We do not know yet whether these two methods are equivalent, and there is 