Instead of describing faces as subsets $\varphi \subset E$, it is sometimes more convenient to choose for each face $\varphi$ an edge $e_\varphi \in \varphi$ and work with the set

$\hat{F}=\{e_\varphi \, | \, \varphi \in F \}\subset E$.

Knowing $e_\varphi$ we can easily recover the other edges that make up the face $\varphi$ by applying $s$ repeatedly to $e_\varphi$.

Similarly, we can choose an outgoing edge for each vertex, a particular orientation for each unoriented edge and work with sets of edges $\hat{V}$ and $\hat{E}$ that are in one-to-one correspondence with $V$ and $E$.

Working with $\hat{E}$, $\hat{F}$ and $\hat{V}$ instead of $E$, $F$ and $V$ can be important for efficient computer implementations. As we will see, it is sometimes also useful for theoretical purposes.