I'm very fond of Ronnie Brown's proof of the Jordan curve theorem using the groupoid version of the van Kampen theorem. I find it easier to understand and reproduce than other comparably elementary proofs, such as Munkres' argument using covering spaces (which can be found in his textbook Topology: a first course). I guess this application isn't really a calculation, as requested, but I hope you like it.
(I guess I should mention that a small gap in the proof is filled in in this short note.)