Here's a cotton cord tied in a continuous six-strand sinnet, making a toroidal knot. This is really a pretty simple knot, and when I try it in silver or gold it will have to have more strands in order to avoid looking too spare and skeletal. It probably wouldn't be practical as either a bracelet or a ring, being much too bulky, but I have not yet ruled that out.
Scroll down for explanations and instructions on tying this knot, or click here for a series of pictures of a similar knot tied in silver wire.
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Lay out two rows of points, in parallel lines, to represent the intervals around the circumference of the knot. The top row and the bottom row actually represent a single line running around the outside of the toroid. Visualize them as the two edges of a zipper, or a row of buttons and button holes, lined up.
Start out with just about anything for a core, but decide ahead of time if you want to be able to remove it or not. I used a separate coil of string, and pulled it out bit by bit. You could tie the knot on a wooden ring, or brass, whatever you like, and keep adding string til it covers it completely.
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The flat diagram wraps around the body of the toroid, with the sides "buttoned" and the ends meeting each other. (Don't actually wrap the paper around it, that'd be a mess. Just keep the diagram handy and look at it when you need to. By the way, ignore the numbers on the inside of this circle -- if you start the string at "1", it will go through the donut and should come out to the outside at point "4", come around and go through again to come out at point "7", and so on. With this one, thirteen points around, the string will come up just short of the starting point on the first pass, further back on the second pass, and meet itself on the third.
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Once you've completed all the turns in one direction (meeting the beginning point) you need to start going around in the other direction. It doesn't matter, when you reverse it, whether you go over the first crossing or under it, but you must alternate them: over one, under one, over one, under one, etc. (The arrows indicate that the line is on top of the other one.) Keep the same angle, come up to the outside points at the same intervals (but from the opposite direction) and continue until you've gone around a full three passes and return to the starting point again.
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Here is how to tie it, in pictures. I started out with the middle of the string, so that half of it can go each way. A bit of dental floss to tie it to the ring is handy, so that you can spot the beginning at any time.
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On the first turn, the line goes around to just before the starting point, making four passes through the center. On the last pass, it could come down on either side of the beginning, but for a 3x13 torus knot it has to come down where you see it, just before the beginning.
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Follow the first turn, passing through on the same side each time, parallel all the way around. If you find it crossing the other strand, you've gone off course.
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On the third turn, you'll be bringing the working end up to the point where the standing part was tied to the ring. I used the bit of floss there to just tie it down temporarily, because now it's time to pick up the other end and begin to work it through.
Note that this is a complete torus knot already, though it won't really stand on its own.
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Now pick up the other end of the string and lead it around in the opposite direction around the ring, and around the body of the ring in the opposite direction as well. It won't matter a heck of a lot whether you go over the first crossing you encounter, or under it, but after that you have to alternate. I just brought it up and to the left, over the end of the first one, as you can see.
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Over one, under one, over one, of course, but you also have to keep count of the times you pass through the center hole. Remember, when you started, it went down through the center four times -- keep it the same. It's easy to lose track at this point, because there is not much structure there to guide you.
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The fourth time through, the line comes out from behind the ring and parallels the beginning again. (Compare with the second picture in this series, almost a mirror image of this one pass of the string.)
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Now you can just follow in parallel with the first pass, but go over wherever it goes under, and under where it goes over. Here you can see the beginning of the third turn to the left, the first pass over the edge and heading through the center, with the over-one, under-one pattern showing clearly.
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I just realized that I shot this picture from the back of the ring, so what you're seeing here is the last pass coming up through the center (remember, we were tucking it down in the other pictures) and going back over the top at the beginning, where the string reverses direction. This is where the "doubling" of the knot begins, leading the string through in exactly the same path to keep the same knot but add bulk to it.
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Here we have the doubling process well under way. Each of the running parts, or free ends, can be passed through parallel to its original course, doubling the path.
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I had to keep going with this, it looked too bare with only two strands in each path. There is still some brass showing, but you can see that it could be continued until the coverage was complete.
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