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This paper mainly studies totally Abelian operators in the context ofanalytic Toeplitz operators on both the Hardy and Bergman space. When thesymbol is a meromorphic function on $\mathbb{C}$, we establish the connectionbetween totally Abelian property of these operators and and geometricproperties of their symbol curves. It is found that winding numbers andmultiplicities of self-intersection of symbol curves play an important role inthis topic. Techniques of group theory, complex analysis, geometry and operatortheory are intrinsic in this paper. As a byproduct, under a mild condition weprovides an affirmative answer to a question raised in \cite{BDU,T1}, and alsoconstruct some examples to show that the answer is negative if the associatedconditions are weakened.

Totally Abelian Toeplitz operators and geometric invariants associated with their symbol curves