It is not so easy to explain because English doesn't have a equivalent for "sin" (as f.x. my language does). But let's say-if we translate "sin" like "its own" (to understand the difference between "sin" and "dets/ dens"), so "sin egen" could be translated as "its very own". It means that the crab has its own plate and he doesn't have to share a plate with somebody else. Or- the plate could be even with crab's name on it.
I think it's wrong to translate ‘sin’ with ‘its own’. The difference between ‘sin’ and ‘dens’ &c. is that ‘sin’ refers back to (usually) the subject of the same sentence, whereas ‘dens’ &c. don't imply such a relation. Maybe you could translate it as ‘its own’ if it were used to contrast with another possessive pronoun, but normally you wouldn't.