Connectedness of the set $\{(z,w)\in S^3\subset \Bbb C^2:z^m+w^n=0\}$ where $m,n\in \Bbb N$

Question

For positive integers $m,n$, let $A_{m,n}$ denote the set $\{(z,w)\in S^3\subset \Bbb C^2:z^m+w^n=0\}$. I am trying to show that this set is connected if and only if $\text{gcd}(m,n)=1$. Also, in this case, I want to compute the length of $A_{m,n}$. But I have no idea how to start for both statements. Can I get a little bit of hints? Thanks in advance.