How to define an angular form with bounded support? In my smooth manifold course, I am asked to define a differential $1$-form $\tau$ on $M = \mathbb{R}^{3} \setminus S^{1}$ with support bounded in $\mathbb{R}^3$ such that $\int_{\gamma} \tau$ can represent the signed crossing number of any closed curve $\gamma : S^{1} \to M$ about the disc $D^{2}$.
I believe that $\tau$ must be related to $\omega = \mathrm{d}(\arctan\frac{z}{\sqrt{x^{2}+y^{2}}-1})$, but how can we modify $\omega$ to obtain a globally smooth $1$-form with bounded support?

It seems that $\int_{\gamma} \tau$ gives rise to linking numbers (see this post and this post).

If you need $\tau$ to have compact support in $\mathbb R^3\setminus S^1$, this is impossible. Here's why. Suppose $\tau$ is such a form, and let $K$ be the support of $\tau$. Because $K$ is a compact subset of $\mathbb R^3$, there is a positive number $\varepsilon$ such that every point of $K$ is at a distance at least $\varepsilon$ from $S^1$. If $\gamma$ is a small circle around $S^1$ of radius less than $\varepsilon$, then $\tau$ is identically zero along the image of $\gamma$, and so $\int_\gamma\tau =0$, although $\gamma$ crosses the disc exactly once.
