  This paper mainly studies totally Abelian operators in the context of
analytic Toeplitz operators on both the Hardy and Bergman space. When the
symbol is a meromorphic function on $\mathbb{C}$, we establish the connection
between totally Abelian property of these operators and and geometric
properties of their symbol curves. It is found that winding numbers and
multiplicities of self-intersection of symbol curves play an important role in
this topic. Techniques of group theory, complex analysis, geometry and operator
theory are intrinsic in this paper. As a byproduct, under a mild condition we
provides an affirmative answer to a question raised in \cite{BDU,T1}, and also
construct some examples to show that the answer is negative if the associated
conditions are weakened.
