Is there some piece of equipment or item that allows you to tie two or more
- strands of rope
- threads of fabric
together? I imagine that (1) and (2) will be different items. If such things do exist, what are they called?
For rope (at least in sailing), this is called splicing (rope splicing) and the primary tool used is a marlinspike. When you're connecting frayed ropes using another piece of (usually thinner) rope this is known as whipping but real splicing involves unlaying the rope (similar to unbraiding) and then laying in new line. See various splices in the diagram below:
There is a process for fabric called reweaving to repair various types of holes but doesn't actually involve "tying". There are several approaches to reweaving discussed here.