I was a commercial fisher in addition to being a homesteader, so I have used a lot of knots. But these are the ones that our family uses daily, are simple, and have lots of uses. For example, the other night a hard frost was suddenly forecast. We drove a few t-posts around sensitive
trees, and the family erected a
shelter in a hurry by flashlight, using string, a few poles, and bedsheets.
sheet bend--When you need to extend your string, knitting yarn, rope, or whatever, attach another piece with this simple knot.
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bowline--Make a loop that takes a lot of weight or pull and yet unties easily. This is used for everything from lifting someone out of a hole (tied under the arms and pulled upward) to towing a car.
figure-eight knot. Another knot that is strong but comes untied easily. Use instead of the ubiquitous overhand knot.
clove hitch--this secures your line to a pole, wire, or other stationary item. When lashing, this is how you attach your string to the pole at the beginning. Because the string never doubles back on itself, but keeps going the same direction throughout, you can make a clove hitch with delicate materials like grass or stems. When I want to tie a small plant like a pepper or flower to a stake, I use a couple blades of long grass twisted together and tied into a clove hitch. Cheap, unobtrusive, and ready to hand. For more demanding jobs like towing a log, the constrictor knot and rolling hitch are heavy-duty variants.
square lashing--the photos in other posts speak for themselves on this. Can be used for substantial buildings as well as light
trellis. I've seen 5-story lashed-pole scaffolds.
square knot--the bow on your sneakers is a slipped square knot (slipped means you have loops so you can pull the ends and untie the knot.) Pull the string all the way through instead of making the loops and you have a square knot. Easy and fast to secure a package or a bundle.
Should never be used to extend a rope or string, as it's not trustworthy for that. Use a sheet bend instead.
trucker's knot--turn any piece of rope into a tie-down instead of buying fancy webbing. This is how you keep a load on your truck or your car roof at 70mph.
We still love the classic Ashley Book of Knots, and find its line drawings to be less distracting than photos, but any source that works for you is good. The only real way to learn a knot is to do it over a few times--it's your fingers that need to really learn it. Arborists are a great idea, and of
course mariners. Brion Toss produced books, and I think a TV series in the 1980's called The Rigger's Apprentice which featured both old-time mariner's knots and newer sailing gear.
I always shake my head when I see the kind of thing people envision for survival gear after the apocalypse. The real necessities are edge tools (knives, drills, axes) and cordage. Then containers and salt. Then with effort and skill you can do the rest.