מסכת כלים

פרק יח משנה ט

A bed,

half of which was stolen (each half consisted of a long board, a short board, and two legs),

or half of it was lost

or brothers split it [in half],

or partners [split it in half],

is tahor (as it is not usable in this state, and is not likely to be reassembled, it is considered to have been broken);

If they were returned

it is susceptible to tumah from that point forward.

[The parts of] a bed can become tamei while connected together (for example, if one of the legs became tamei while connected to one of the boards, they'd both be tamei, as they are considered a single unit),

and can be purified while connected (i.e. they can be immersed while still connected, again because it is considered a single unit),

this is the opinion of Rabbi Eliezer,

and the Sages say

they become tamei in parts

and [must be] purified in parts (unless in a complete bed, bed components will be viewed as separate items even when connected to each other)

פרק יט משנה א

One who takes apart a bed (that was tamei)

to immerse it (in a mikvah)

and one who touched its ropes (these were the ropes that formed the bed's surface, had been a part of the the bed when it became tamei, and so were tamei as well),

is tahor. (There different interpretations of this part of the mishnah.  Some say it is referring to a bed that had already been partially disassembled when this person touched it, and that in such a disassembled state, its parts could not confer tumah; the novelty being that since the intent was to reassemble it after immersion, one might have thought it would still impart tumah.  Others say that the person who took apart the bed will become tamei, and the ruling of the mishnah is only with regards to a person who touched the ropes after the bed had been disassembled, as the ropes are not reassembled as easily as the other parts of the bed, they lose their ability to impart tumah.)

The rope

from when is it considered connected to the bed?

from when three boxes are woven in it (in the network of ropes that would be woven to form the sleeping surface; some say this means three squares of rope of that weave , others interpret this as meaning that three rows of the weave were completed).

And (if the rope was not long enough t o complete the weave for the entire bed, and so was knotted to a second rope, which one planned to then continue the weave with,) one who touches from the knot (which joined the first and second rope) and inward (ie on the first, already woven, rope)

is tamei,

from the knot and outwards (on the second, unwoven rope; this does not include the knot itself)

is tahor;

The strands of the knot (the running end of the rope that comes out of the knot),

one who touches the necessary part (that is the amount of the running end that is necessary to be able to tighten the knot, and to keep it from untying)

is tamei;

and how much is necessary?

Rabbi Yehuda says

three fingerswidths.