Now proceed by induction. Suppose that is a path around n points such that if you remove the hole at any one of those n points then the path becomes null-homotopic. Define If you remove one of the points (for ) then becomes null-homotopic and so becomes . And if you remove then becomes 1 and so becomes . That completes the inductive proof that has the required property.
Once you have that formula, it's easy in principle to draw a geometric picture of the loop for example. But the complexity of the loops increases alarmingly as the number of holes increases. Even for n=3 it will look fairly messy.