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abstract: 'The Gauss self-linking integral of an unframed knot is not a knot invariant, but it can be turned into an isotopy invariant by adding a correction term which requires adding extra structure to the knot. We collect the different definitions/theorems/proofs concerning this correction term, most of which are well-known (at least as folklore) and put everything together in an accessible format. We then show simply and elegantly how these approaches coincide.'
address: 'Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502 JAPAN'
author:
- Daniel Moskovich
date: '04.03.21'
title: 'Framing and the Self-Linking Integral'
---

Introduction
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In 1833 Carl Friedrich Gauss, in his investigation of electromagnetic theory, discovered an integral formula for the linking number of two space curves. If $\gamma_{0}$ and $\gamma_{1}$ are disjoint embeddings of $S^{1}$ into $S^{3}$— i.e two disjoint space curves, and if $\Phi:S^{1}\times
S^{1}\rightarrow S^{2}$ is the map that assigns to each $(x,y)\in
S^{1}\times S^{1}$ the unit vector from $\gamma_{0}(x)$ to $\gamma_{1}(y)$, with $\omega$ defined as the volume form on $S^{2}$, the Gauss integral is $$\frac{1}{4\pi}\int\limits_{S^{1}\times S^{1}}\Phi^{*}\omega$$ It is natural to ask what happens if we take $\gamma_{0}$ to equal $\gamma_{1}$, in other words if we want to find a knot invariant 