Define the algebraic intersection number between two simple closed curves $\alpha$ and $\beta$, denoted by $\#(\alpha,\beta)$, in a surface by the sum of all indices of the points of intersections between $\alpha$ and $\beta$, where the index is +1 if the orientation of the pair $(\alpha,\beta)$ agrees with the orientation of the surface and -1 otherwise.
Let $\alpha_1,\alpha_2, \beta_1, \beta_2$ be simple closed curves in a surface $S$. Suppose $[\alpha_1]=[\alpha_2]$ and $[\beta_1]=[\beta_2]$ in the first homology group of $S$, $H_1(S)$. Then we obtain $\#(\alpha_1-\alpha_2,\beta_1-\beta_2)=0$. Does the latter imply that $\#(\alpha_1,\beta_1)-\#(\alpha_2,\beta_2)=0$?