The problem: Consider a family of projective curves $X_t$ in $\mathbb{P}^2$ given by the equation $$x^3+y^3+z^3-txyz=0$$ with homogeneous coordinates $[x:y:z]$. Let $U=\{t\in\mathbb{C}:|t-3|=\delta\}$ for a small positive number $\delta$. Try to construct a family of loops $L_t:\mathbb{S}^1\to X_t$ for points $t\in U$, such that each loop $L_t$ represents a primitive homology class in $H_1(X_t;\mathbb{Z})$. (An element $\alpha\in H_1(X_t;\mathbb{Z})$ is said to be primitive if it cannot be written as $\alpha=n\alpha'$ for some $\alpha'\in H_1(X_t;\mathbb{Z})$ and $n\ge2$.)
I know that in the case of $t=3$, the curve $X_3$ degenerates to a singular curve. So in my problem $X_t$ is obtained by perturbating the singular curve a bit, resulting in a smooth cubic curve of genus $1$.
It is persuasive that there exists such a family of loops (meridians?) representing homology classes, which remain primitive as the parameter $t$ varies. But I cannot give an explicit expression of this family of loops.
Any idea or hint is welcome! It would be better if someone could write down an explicit expression :)