I saw the definition of rotation number in this post about differential forms. Here it is for convenience.
A complex differential $k$-form $\omega$ has rotation number $p-q$ if
$$\omega (e^{i\theta} v_1, \dots, e^{i\theta} v_k) = e^{(p-q)i \theta} \omega (v_1, \dots, v_k),$$
with $p+q=k$.
I am wondering what the intuitive explanation for this is. For instance, suppose we have a one form in $\mathbb{R}^3$, $\alpha = a \, dx + b \, dy + c \, dz$. Then we can (loosely) think of $\alpha$ as the vector field:
$$\mathbf{F} = a \, \hat{i} + b \, \hat{j} + c \, \hat{z}, $$
(given that the line integral of the field $\mathbf{F}$ over a curve $C$ is the equal to the integral of the differential form $\alpha$ over $C$.)
Here is my question: If $\alpha$ has rotation number zero, does it say anything about the physical nature of the vector field $\mathbf{F}$? For example, does the vector field not change under rotation, or does it have no curl, etc.? I'm looking for a way to make the abstract "rotation number" definition more visual and tangible. Any help would be appreciated!