[almostgotcaught]

i mean calling arnold's proof actually topological is probably a stretch (the "deep connection" you're talking about). it doesn't really use any topological facts about either the loops or the embedding space. continuity isn't really required i don't think? i should've said contiguity instead. it's just a very very nice model for the theory (in the sense of model theory) that lends itself to immediate visualization.