Mikvaot, Chapter Ten, Mishnah Five
1) Any handles of vessels which are too long and which will be cut short, need only be immersed up to the point of their proper measure.
a) Rabbi Judah says: [they are unclean] until the whole of them is immersed.
2) The chain of a large bucket, to the length of four handbreadths, and a small bucket, to the length of ten handbreadths, and they need only be immersed up to the point of their proper measure.
a) Rabbi Tarfon says: it is not clean unless the whole of the chain-ring is immersed.
3) The rope bound to a basket is not counted as a connection unless it has been sewn on.
Section one: If the handle of a vessel is too long and is supposed to be cut off, but has not yet been cut off, it needs to be immersed only up to the point at which it will be cut off. We consider the remainder of the handle as if it had already been cut off.
Rabbi Judah says that since the extra part of the handle is connected to the part that will remain, he must immerse the whole thing.
Section two: Up until four cubits, the chain of a large bucket is considered to be its handle. But longer than that, it is not necessary for the bucket and is not considered to be part of it. Therefore, it doesn’t need to be immersed beyond four cubits. The small bucket must be immersed up until 10 handbreadths because it is lowered using this chain into the well. (See Kelim 14:3).
Rabbi Tarfon adds that if the proper measure (4 or 10) ended in the middle of a chain ring, then the whole ring must be immersed. A bit nit-picky, but no one’s asking my opinion, are they?
Section three: A rope attached to a basket is not considered to be connected to the basket, so it need not be immersed with the basket.