I am currently learning the surface code and I wanted to know if there is an easy and intuitive way to find out all the equivalent possible definitions for the $X_L$ (Pauli-X logical operator).
In the following images, the black qubits surrounded by green are the $Z$ stabilizers, the black qubits surrounded by yellow the $X$ stablizers and the white circles are data qubits. The red cross means that I apply an $X$ operator on the associated data qubit.
I originally thought that as long as you have a crossing-line composed of $X$ Pauli then you have a logical $X$ operator defined. But I think it is not correct given my third example.
First example: this is the "standard" way to define the logical Pauli $X$:
I could also do the following:
But for instance, this wouldn't work:
Indeed, if I am correct in this last example the $Z$ stabilizers of the third line, second and fourth column will not commute with the operator I applied.
Hence, is there a nice geometrical and intuitive interpretation to define logical operators or I have to check for each attempt to build a logical $X$ (which is crossing the surface with $X$ operators) if all the stabilizers are commuting ?