Injectivity of the simple closed curves under geometric intersection number Let $\Sigma$ be a closed surface of genus $g\geq 2$ and $\mathcal{C}$ be the set of  all free homotopy classes of simple closed curves in $\Sigma$. Define $i:\mathcal{C}\rightarrow \mathbb{R}^{\mathcal{C}}$ by $$i(x)(y)=i(x,y)$$ for $x,y\in \mathcal{C}$ where $i(x,y)$ is the geometric intersection number.  
Q) Does there exist a finite subset $K\subset\mathcal{C}$ such that $i:\mathcal{C}\rightarrow R^{K}$ is injective? 
 A: Yes, there is a collection of $9g - 9$ curves that suffices.  See Expose Six and Appendix C of "Thurston’s Work on Surfaces" edited by Fathi, Laudenbach, and Poenaru.

Here is a very short sketch - the details are complicated.  Suppose that $\Sigma$ is the given surface of genus $g$.  Let $P = \{\gamma_i\}$ be a pants decomposition of $\Sigma$.  We first recall the Fenchel–Nielsen coordinates.  Suppose that $\sigma$ is a hyperbolic metric on $\Sigma$.  For every $\gamma_i \in P$ we then have 


*

*a hyperbolic length $\rm{len}(\gamma_i, \sigma)$ and

*a twist parameter $\rm{tw}(\gamma_i, \sigma)$.


The former is a positive real number and the second is a signed real number.  Care must be taken to understand the twist as a signed real number and not just as a circle valued number. We can recover the twist from length data as follows.  For each $i$, pick dual curves $\alpha_i$ and $\beta_i$ so that the triple of curves $\{\alpha_i, \beta_i, \gamma_i\}$ meet each other non-trivially and minimally, and meet the rest of $P$ minimally.  Then the lengths of these $9g - 9$ curves recover the twist parameters. 
We now can discuss the original question.  Again we have $\Sigma$ and $P$.  We now recall the Dehn-Thurston coordinates.  Suppose that $\delta$ is a simple closed curve.  For every $\gamma_i \in P$ we then have  


*

*an intersection number $\rm{i}(\gamma_i, \delta)$ and

*a twist parameter $\rm{tw}(\gamma_i, \delta)$.


The former is a non-negative integer and the second is an integer.  Care must be taken to understand the twist.  It can be defined using from intersection data with dual curves, chosen as above. 
