Assume $V$ is a 3D smooth divergenceless ($\nabla \cdot V = 0$) vector field which vanishes outside some finite region $\mathcal{R}$. We can intuitively imagine such a field as a bunch of integral curves looping around one another. Is there an invariant associated to such a field which tells us "how many" integral curves of $V$ there are in $\mathcal{R}$.
For example, if $\mathcal{R}$ has the topology of a torus, and all the integral curves of $V$ circle inside the tube, we can calculate the flux of $V$ at any cross section of the torus and this would give us "the number of integral curves of $V$". Can this be generalized for an arbitrary, but well-behaved, region $\mathcal{R}$ and divergence-less vector field $V$?