I need to travel from Coimbatore (India) to Stockholm Arlanda (Sweden). I have a Swedish residence permit.
Could you help me find a route across the world for which I would not need any transit visa?
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This will not answer your question directly, but the following will tell you that you can go through any Schengen country on your way. That should open up a long list of possible travel routes from India to Sweden.
A PUT (Permanent uppehållstillstånd, Permanent Residence Permit) for Sweden (or any other Schengen country) works the same as a Schengen Visa. That means you can travel through any Schengen country on your way to/from Sweden, even take a vacation in say Norway.
Foreign citizens with residence permits in a Schengen country, and whose identities have been properly established, may also travel freely in the zone for three months.
From the Swedish migration office website.