Consider the closed Surface $S$ represented by the word:
$a_{1}\cdots a_{n} a_{n+1} \cdots a_{2n} a_{1} \cdots a_{n} a^{-1}_{n+1} \cdots a^{-1}_{2n}$
Use word rules to convert this sequence into a surface that you recognise and state what that surface $S$ is.
Comments:
I'm aware of how to manipulate word rules to get to some more identifiable surface, but with this sequence I struggle with determining what I might get for some $n$. For instance, what does $a_{2n}$ exactly mean? I'm unsure if this term represents a string of 2n edges, or if it means the $2_{n}$th edge. Any guidance on how to classify this surface given the word rules would be appreciated.
It's the ($2n$)th edge. So for example, with $n=3$, we have $$a_1a_2a_3a_4a_5a_6a_1a_2a_3a_4^{-1}a_5^{-1}a_6^{-1}.$$