Keilim, Chapter 19, Mishnah 1. Here the Mishnah takes up the bed together with the ropes strung through it, asking which of a bed's components are regarded as one item with it when the laws of tumah are applied. To picture the discussion properly, imagine how beds were built then: four legs supported a rectangular frame, made of two short boards at the head and at the foot and two long boards along the sides, and cords were run through it in both directions, crossing to form a mesh that held the bedding.
One rule underlies everything the Mishnah says here. Suppose a bed became an av hatum'ah and was afterwards taken to pieces. What remains is no longer one object at all, and a heap of separate boards and cords is useless as a bed. Having lost its function, it also loses its tumah. But if the dismantling was done with the plan of assembling the bed again, the original tumah settles back on it as soon as the pieces are joined. Precisely for this reason, an owner who wants his bed to be pure takes it apart and brings each component to the mikveh on its own. When all the pieces have been immersed, the bed that is put back together is tahor, and the tumah it once had does not return.
Touching the parts while they lie apart
Our Mishnah asks about the person handling the bed during this process: "hamefarek et hamitah lehatbilah", one who takes a bed apart in order to immerse it, since the bed as a whole is too large to be brought into a mikveh intact and must be separated into its pieces; "vehanogea bachavalin", and likewise one who touches the ropes that had been part of the tamei bed and are now detached; "tahor", he remains pure.
Although the bed itself had been tamei, the loose parts carry no tumah at this moment. They cannot be used as a bed in their present state, so there is nothing tamei for the person to contract. The same applies to the ropes: once the bed has been taken apart, the ropes are no longer a component of a functioning keli, and one who touches them stays tahor.
When does a rope become part of the bed?
The Mishnah now turns to cords in the opposite situation: a rope in the process of being threaded into the frame. How far must the weaving proceed before the rope is reckoned as attached, so that someone touching it would receive tumah from a tamei bed? The Mishnah's question is "Hachevel mei'ematai chibur lamitah?", from what point is a rope joined to the bed, and it answers "Misheyisrag bah shelosha batim", from the moment three rows of it have been woven in.
Once the rope has been woven to that extent, the connection is established. From then on, if the bed becomes tamei, it is not only the woven portion of the rope that is tamei, but the entire length of the rope, including the part still hanging loose, since that remaining length is destined to be strung into the bed and treated as part of it.