algebraic-topologyfundamental-groups
I think that very similar questions have been asked elsewhere on MSE, so if there is nothing new here I apologize.
I know that the fundamental group of the $n$-punctured torus is the free group $F_{n+1}$, as we can see by considering the punctured torus to be the wedge sum of of $n+1$ circles. My question is, what, geometrically, are the generators of the fundamental group?
We have two representations of the Klein bottle as a fundamental polygon. The first is:
And the second is formed by cutting this polygon across the diagonal, flipping one piece and reattaching it to give essentially two real projective planes glued together:
You should be able to see that as CW complexes and a 2-cell attached according to the diagram, these both form the Klein bottle with non-orientable genus 2.
Removing a point from this 2-cell produces a space that deformation rectacts onto the 1-skeleton, which in both cases obviously forms the wedge sum of two circles and the fundamental group is $\Bbb{Z} * \Bbb{Z}$.
Let's see if we can develop some sort of physical intuition for this. If the point (or by deformation, hole) we remove is in the right place, we can embed this in $\Bbb{R}^3$ to get a physical intuition.
Which then forms 
And you can see rather easily that this deforms to:
Which obviously has the fundamental group of $\Bbb{Z} * \Bbb{Z}$, as this deformation retracts onto $S^1 \vee S^1$.
Here is the fundamental polygon of the torus.
The punctured torus simply has a point $P$ removed. It's best to think of $P$ having been removed from the gray area. Your proof claims any lift of a generator of $\pi_1(U\cap V)$ is a loop (with winding number 1) around $P$. Recalling that elements of fundamental groups are invariant under homotopies, we can retract this loop to run around the boundary. Starting at the bottom right of our diagram and traveling counterclockwise, we see this look $ABA^{-1}B^{-1}$.
Best Answer
There is, of course, no canonical choice of generators, but here is perhaps the "geometrically simplest/most obvious" choice.
$n$ punctured torus with $n+1$ loops" />
Notice that if we filled in each of the punctures, each of $a_1$ to $a_n$ would be homotopic to the same loop $b$ and we recover the standard set of generators of the fundamental group of a Torus $\langle a=a_0,b\mid[a,b]\rangle$.