Identify the Euler characteristic of the edge word $ abc^{-1}b^{-1}da^{-1}d^{-1} c $.
The Euler characteristic is $$ X=V-E+F$$ where $V$, $E$ and $F$ are the vertices, edges and faces respectively.
The solution to the problem is given below.
I do not understand how to find the number of vertices from the polygon model or edge word
l.
You ask how to find the number of vertices for this edge word.
We begin with $8$ distinct vertices. Identifying the two edges labeled $a$ eliminates $2$ vertices (since two vertex pairs are glued together). Then the identification of the two edges labeled $d$ identifies these two vertices (look at the $a$ edge in the lower right), so we subtract $1$ again.
The two vertices at the $c$ edge in the upper right have already been identified, so gluing the two $c$ edges eliminates $2$ more vertices. Finally, we see in the same way that gluing the $b$ edges eliminates $2$ more.
In this way, we see that all $7$ vertices not equal to the one in the lower right, where $a$ and $d$ meet, are identified to this vertex. So all vertices, after identification, become one vertex.