I'm trying to get some intuition about the multiplicity of a zero locus (not necessarily discrete!) for a vector field $v$.
Let me give you the context: Let $M$ be a closed, connected, oriented $3$-manifold. Let $\xi$ be a contact structure on it, (it's not really relevant here, but think of it as a rank $2$-subbundle of $TM$ with some properties) Let $v$ be a vector field in $\xi$. Clearly by some transversality argument we can assume that the zero locus of such vector field is a $1$-submanifold of $M$ (since my vector field is in $\xi$, and not just in $TM$, therefore I don't have all the space I need for having discrete zeroes). According to Gompf's $4$-manifolds and Kirby Calculus, we can speak about multiplicity of such zero locus:
The equality $$2[\gamma]=PDc_1(\xi)=PDe(\xi)$$ should come from the fact that I can push the zero section of $\xi$ in the $v$ direction and using that to compute $[Z]=PDe(\xi)$, clearly I would have intersection on $\gamma$, being $v=0$ there. Therefore I'm trying to build some intuition about what kind of definition of multiplicity should I use in order to ensure that $2[\gamma]=[Z]$ holds.
If I look (locally) at $Z$, what should I see? something like $2$-times winding of $v$?