the ‘point-splitting regularization integral’. Tze [@Tze] quotes a ‘simplified approach’ which he credits to Anshelevich, as quoted in an article about the twisting of strands of DNA [@KaV] (!), which leads to the conclusion that the correction term must $\frac{1}{2\pi}$ times the total holonomy of the curve. But the same result may be obtained in a more elementary way by making use of a technique we have already used— that of a ‘swaddling map’. This will also help us to visualize why and when the two constructions for the error term of the self-linking integral will coincide.

Two Ways of Looking at the Same Thing
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### Don’t take limits- compactify!

Rather than thinking of limits of integrals, let’s compactify the space of pairs $$\mathbb{L}^{\overrightarrow{n}}(S^{1}):=\{(t, t+
\varepsilon\cdot n(t))\mid t\in \text{\{a closed space curve\}},
0<\varepsilon\in\mathbb{R}\}$$ by pasting something onto the boundary $\mathbb{L}^{\overrightarrow{n}}_{0}(S^{1}):=
\overline{\mathbb{L}}^{\overrightarrow{n}}(S^{1})\mid_{\varepsilon=0}$, where the overline denotes topological closure. Our problem, as usual, is that $\Phi(x,x)$ is not defined, and what this second approach gives is a way of defining it via a limit which keeps track of the information which is relevant to the Gauss integral— direction— and thus tells us what it wants $\Phi(x,x)$ to be.

Thus, the space 