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

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.