Given points (or holes) , write for (the homotopy class of) a clockwise loop around the point , and write 1 for the homotopy class of a null-homotopic loop. The loop in the example linked to above is given by (where the inverse denotes the corresponding anticlockwise loop, of course). If you remove one of the points then the corresponding loop becomes equal to 1. So for example if you remove the hole at then the path becomes .

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.