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?
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.