Let $C$ be the plane curve in $\mathbb{C}^2$ defined as $\{x²-y^3=0 \}$. The fundamental group of $\mathbb{C}^2 \backslash C$ is the same of the trefoil knot : $\langle a_0,a_1 \; : \; a_0a_1a_0= a_1a_0a_1 \rangle$, see for example M.Oka.
Let $\Omega \subset \mathbb{C}^2$ be such that the fundamental group of $\Omega \backslash C = \langle a \rangle$ isomorphic to $\bf{Z}$.
Here is my question. Can I find a loop $a_1 \subset \mathbb{C}^2\backslash{C}$ such that $\pi_1(\mathbb{C}^2\backslash C)$ is isomorphic to $\langle a,a_1 \; : \; aa_1a=a_1aa_1\rangle$ ?
Thanks.