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The nomenclature I use to describe figures is based on the style developed by the International String Figures Association. For the openings and extensions, I have generally adhered to the names that are most commonly used in the string-figure literature. 
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Different string figures work best with loops of different sizes. Suggested string length is measured in spans, where one span is defined as your height, or the distance between your fingertips when both arms are outstretched to the sides. Thus, a "1 span loop" is made by joining the ends of a piece of string whose length is the same as your height, while a "2 span loop" is made from a string twice that length.
1 = thumb, 2 = index, 3 = middle, 4 = ring, 5 = little, H = hand
L before finger designation = left (e.g. L1 = left thumb)
R before finger designation = right (e.g. R1 = right thumb)
n after finger designation = near
f after finger designation = far
u before finger designation = upper (distal)
l before finger designation = lower (proximal)
distal = nearest the fingertip, regardless of the finger’s orientation (distally = from fingertip side)
proximal = nearest the base of the finger, regardless of the finger’s orientation (proximally = from the direction of the base of the finger)
TV = transverse, running straight from one hand to the other
palmar = crossing the palm of the hand
dorsal = crossing the back of a finger or hand
When there is no L or R designation before before a finger number, the action is performed simultaneously on both hands unless otherwise stated.
pick up = retrieve a string using the nail side of a finger
hook up = retrieve a string using the fingerprint side of a finger, then straighten
the finger by rotating it a half turn towards you or away from you as indicated
hook down = retrieve a string using the fingerprint side of a finger, securing
it in the crook of a bent finger. Often the finger remains in this position with the
fingertip touching the palm
transfer = move a loop from one finger to another (e.g. "proximally transfer 5
loop to 2" means "on each hand, insert index finger into the loop on the little finger,
entering it from the knuckle side, then withdraw the little finger")
exchange = swap the loops encircling the same finger of each hand so that one loop
passes through the other (e.g. "exchange 2 loops, inserting fingers distally, passing R
loop through L" means "transfer right index loop to left index, inserting left index
from the fingertip side; insert right index into the proximal left index loop (nearest
the base of the finger), entering the loop from the fingertip side, then lift it over
the distal left index loop and off the finger to complete the exchange"
release = allow the loop to slip off the finger
extend = absorb the slack by separating the hands
> = rotate the indicated loop a half turn (180°) away from you
< = rotate the indicated loop a half turn (180°) towards you
>> = rotate the indicated loop a full turn (360°) away from you
<< = rotate the indicated loop a full turn (360°) towards you
Navaho = With the help of the opposite hand, lift the proximal loop(s) over the distal loop and release it.
Double Navaho = Using the opposite hand, shift the proximal loop over the distal loop towards the tip of the finger, then lift the new proximal loop over the first loop and release it.
Reverse Navaho = Using the opposite hand, reach down through the distal loop to grasp the proximal loop; lift both loops off the finger, releasing the distal loop which slides over the other loop; replace the held back onto the finger.
Index Twist = At the beginning of this movement there will be (at least) one 2 loop and usually two 1 loops. Proceed as follows: Insert 3 towards you (proximally) through 2 loops, then catch proximal 1n (which will be TV) between the 2 and 3 fingertips and pull it back through the 2 loop, placing the former proximal 1n around 2 by rotating the fingers away from you. You will then either Navaho this string off the thumbs, or drop the thumb loops completely, as specified in the instructions for the figure in question. On rare occasions you pick up a string other than proximal 1n; this will also be specified in the instructions.
R Katilluik = Proximally insert R1 into L1 loop; withdraw L1 from its loop and insert it proximally into both R1 loops; proximally insert 1 into 2 loop picking up 2n, lift double proximal 1 loop over the distal 1 loop and release it; end by releasing 2 loop.
L Katilluik = Same as R Katilluik but initiated by L1 (switch R1 and L1 in the above description)
The techniques described below are designed to optimally display a string figure, and are always the very last step in constructing a figure. For these techniques to work, there must be loops on only 1 and 5, plus 1n and 5f must be transverse (the power lift is an exception - see below).
This is probably the most widely-used extension, especially among peoples of the Pacific islands (it's named after the Caroline Islands). Step 4 step is called the Caroline extension (or or sometimes the Pindiki stroke), but I have included the preceeding arrangement steps and called the whole thing "Caroline ending".
This my favourite extension. It creates a wider display than other methods, is easier to do, and allows the figure to be arranged easily. The technique was used to display a New Zealand Maori figure called Kotiro-Punarua which was collected in 1920 by Johannes Andersen, and that is still the only published account of this technique that I have seen.
This extension was invented in 1969's by Steve Newkirk for the purpose of displaying his own "engineered" figures, but the technique can be usefully applied to a wide variety of figures. Note that this will display figures upside-down compared to the other extensions.
James Murphy invented this very useful technique. It is similar in effect to the Caroline ending, but quicker and simpler to do, and to undo. But the really great thing about the power lift is that you can apply it to figures involving any number of loops, as long at the strings at 1n and 5f are transverse. This means it is excellent for checking progress part-way through a complex figure.