Knots inside loops

Question

Reading about practical camping, sailing... knots, you find some of them can be done without having access to the rope's ends (if you don't require it to be attached to another loop, ring, etc.). That is, they can be performed at some point of a closed loop or an endless rope, as you don't need to manipulate its ends (what they refer as to perform the knot in a bight [see section]). It seems, with no knowledge of topology, that the criteria to classify knots into this or the opposite categories must be something basic related to parity or some easy to check condition, but I don't manage to find the result, probably due to my lack of topology vocabulary. ¿Can anyone shed some light on this?