Geometric/Algebraic intersection numbers of curves on surfaces I have the following problem, and struggling to find some references.
Suppose I start with a homology class of a curve on a closed genus $g$ surface $$h=(a_{1},b_{1},\dots,a_{g},b_{g})\in H_{1}(\Sigma_{g};\mathbb{Z}).$$
If $h$ is primitive, I can find a simple closed curve $\gamma$ such that $[\gamma] = h$. Now I want to find another disjoint simple closed curve $\gamma'$ so that $\gamma$ and $\gamma'$ don't intersect. We have that $[\gamma'] = (a_{1}',b_{1}',\dots,a_{g}',b_{g}')\in H_{1}(\Sigma_{g};\mathbb{Z})$ for some $a_{i}',b_{i}'\in\mathbb{Z}$. 
What restrictions can we place on the $a_{i}'$ and $b_{i}'$ to ensure that $\gamma'$ is disjoint from $\gamma$?
Clearly we would require that we have for the algebraic intersection number $\hat{i}([\gamma],[\gamma'])=0$, but it seems that the algebraic intersection number is not easily computable. I think I found the apocryphal formula
$$\hat{i}((a_{1},b_{1},\dots,a_{g},b_{g}),(a'_{1},b'_{1},\dots,a'_{g},b'_{g})) = \sum_{i=1}^{g} a_{i}b'_{i}-a'_{i}b_{i},$$
but can't find a proof.
Any references, advice, or solutions would be greatly appreciated!
 A: Collecting my previous comments here. The formula you give for the algebraic intersection number is correct, and points to the fact that the intersection number is a symplectic form on $H_1(\Sigma_g)$ (with $\mathbb{R}$-coefficients if you want to make it a vector space). A reference for this is Farb and Margalit's A Primer on Mapping Class Groups, Chapter 6. 
You said that algebraic intersection number zero is necessary for geometric intersection number zero, which is certainly true. However it is not sufficient. Just take this example on the genus 3 surface (or any higher genus):

The trouble is that specifying a homology class fails to really specify a homotopy class of simple closed curve. For instance if you chose $\gamma'$ to go above rather than below the middle hole in this picture, you'd have a homologous curve which is now disjoint from $\gamma$.
However in the genus $\leq 2$ case, I believe $\hat i=0$ does imply $i=0$ for simple closed curves representing primitive homology classes. This is easy on the torus, since simple closed curves here in minimal position intersect with the same orientation at all points. I haven't been able to come up with a proof yet in the genus two case beyond lack of counterexample.
