Reading Bachman's "A Geometric Approach to Differential Forms", in section 7.8.1 about the Lining Number invariant, I have stumbled upon the following assertion.
Let the knot $K$ be defined as a (continuous and differentiable) map from unit circle $S^1$ to 3d Eucledian space $\mathbb{R}^3$, so:
$K: S^1 \to \mathbb{R}^3$
i.e. $K$ is a closed curve in 3d space parametrized by a single real variable.
Let $U=\mathbb{R}^3-K$, i.e. it is the 3d space without the points associated with the knot (curve).
Then there always exists a map: $A:U\to S^1$.
Unfortunatelly author then says that the proof is outside the scope of the book, and I am not sufficiently knowledgeable to know where to find the proof. Can anyone please suggest a good reference for a keen amateur?
Paragraph from the book:
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The linking number of a two-component link is precisely the analogous measure that you get when you treat one of the components as the z-axis and the other as the 1-chain over which we integrate. Given any knot $K$ in $\mathbb{R}^3$, there is a function $A: \mathbb{R}^3 − K \to S^1$. (The existence of such a function is, unfortunately, beyond the scope of this book.) If we think of A as a 0-form, then we can differentiate it to get a 1-form on $\mathbb{R}^3 − K$. This 1-form is precisely what we can integrate over a second knot to measure how many times it “links” with $K$. Interestingly, there is also a point $p \in S^1$ such that $A^{-1}(p)$ is a surface whose boundary is $K$. Just as before, the linking number can also be computed just by appropriately counting the number of intersections with this surface.
"""