pieces onto it in order to “force” the Gauss integral to converge and to give us an invariant. There is another way to approach the problem however. We already know that the linking number is a link invariant— well then, let’s pretend that our knot is a link! If we take two copies of our knot that are only an $\varepsilon$ apart, and then see what happens when $\varepsilon$ goes to zero, we may utilize the known invariance of the linking number for links in order to directly conjure up an invariant for knots.

In the approach we are now taking, we have one ‘stationary’ curve, which we shall label $\gamma_{0}$, for which we choose a smooth framing $n(t)$. We then take the curve obtained made up of points $t+\varepsilon\cdot n(t)$ for $t\in \gamma_{0}$, with $0<\varepsilon\in \mathbb{R}$, which we shall denote $\gamma_{\varepsilon}$. Now we let the ‘mobile’ $\gamma_{1}$ descent towards the ‘stationary’ $\gamma_{0}$. What we want to know is what happens the self-linking integral when they touch.

The classical approach here is to take the limit as $\varepsilon
\rightarrow 0$ of the Gauss integral, which involves writing and partially calculating an explicit integral. This leads us to what physicists call 