What Knots To Like

String figures have been fascinating people all over the world since time immemorial. They have traditionally been made with a circle of string, essentially a long thin flexible torus which is manipulated on the hands to make pleasing figures. I have been playing with my circles of string since the middle of 1979 when i was called on to teach math and science to a class of 10-11 year old gifted male students in a Saturday program. In order to intrigue them and give the squirmy young men something to do i used Jayne’s book on string figures to discover how to do the ten men figure and worked out a series of permutations of the basic weaving moves to make an enjoyable exploration of the possible figures for my charges. This experience was a revelation for me and i have never looked back.

The torus can also be described as the null knot. This is to say that it is not a knot at all but a torus. The simplest knot is the trefoil which is a continuous string without end and which cannot be manipulated into a simple circle. A good illustration used by mathematicians is this:

As you can see there are two flavors of the simple trefoil knot which you should expect for we live in a chiral world. The left and right hands do not fit on each other both palm down. The thumbs stick out either side. But place palms together and they nest quite symmetrically.

You might ask yourself looking at the two figures, how could one form a figure placing the string on both hands and manipulating the various strings running between the fingers. The best way to do this would be to rearrange the knotted loop of string until it had the following orientation:

The top trefoil knot was formed by holding the unfused ends of the string in the hands with the loop hanging down and the right hand string was placed over the left hand string and wrapped around and pulled through the circle of string formed between the hands. This is the right overhand knot. Then the two ends were fused. Note that the bottom left string of the figure goes over then under then over the top left hand string. i will call this loom the over under over loom or simply ouo.

The bottom trefoil knot was formed with the left hand string going over the right hand string then pulled back through the circle before fusing the ends. This is the left overhand knot and will be notated as the under over under or uou loom.

The top mathematical representation of the trefoil knot above is the ouo (read from the bottom left string entering the wrapped portion of the torus) ready for string figures and the bottom mathematical trefoil knot above is the uou (again the bottom left string crossing read left to right).

Now we place the top loop on the thumbs and the bottom loop on the little fingers and form a third loop in two ways. The left thumb can go down into the right index loop near the right hand and curl back lifting the far right index string until the right thumb can be inserted from below into the newly forming loop and the hands separated until a three loop position is formed on the hands. We can also easily have used the right thumb to pick up the near left index string and the left thumb inserted from below to form the third (thumb loop).

In this figure the left ouo loom had the left hand form third loop and the uou loom had the right hand form the third loop.

Now we are ready to explain how to form the position from which all the figures of this first part of the knot article will be formed.

Either the ouo or uou loom is placed on the index and little fingers with a transversal as the near thumb string and a transversal as the far little finger string

Either the left hand or the right hand leads to form the third loop on the thumbs

the figure is rolled on the hands

the bottom is fixed

both index loops are rotated – 1/2

So there are four positions we have arrived at. There is a nascent bottom of a figure with an ouo or uou double wrap suspended above a bottom transversal with a double wrap on either side of a figure yet to be formed. There is a transversal loop on the thumbs and two loops lifted from the two index fingers. One of these loops has both strings go over the top string of the other loop and the other loop has one string pass over and one string pass below the other loops top string. For much of this paper one of these four positions will be the starting point for making a figure. They are notated simply as:

ouo left
ouo right
uou left
uou right

Now let’s look at what we can make.

For the next eight figures one more step has to be added before the chopstick heart move. For the left hand continuations, the left index loop is taken with the left thumb and index and pushed down through the center hole to be taken up again by the left index so there is an uou wrap between the index fingers.

For the two right hand led continuations, the right index loop is pushed down and through the center hole to form a double wrap

The resulting two double wraps can be envisioned as a knot and the following shows the results:

ouo left (left index down and up through center hole) apparent granny knot
ouo right (right index down and up through center hole) apparent square knot
uou left (left index down and up through center hole) apparent square knot
uou right (right index down and up through center hole) apparent granny knot

Now let’s investigate patterns by making eight figures.

ouo left, left index pushed down and up through, ch right

Note the knot is a square knot and the center heart is free to slip through the knot in either direction. This is a stable figure under tension.

ouo left, left index pushed down and up through, ch left

Note the knot is a granny knot and the center heart cannot slide out in either direction. The center heart also must be arranged and suffers intensionality.

ouo right, right index pushed down and up through, ch right

Note the knot is a square knot and the heart cannot slide out in either direction. The center heart must be arranged and suffers intensionality.

ouo right, right index pushed down and up through, ch left

Note the knot is a granny knot and the heart can slide in either direction. This is a stable figure under tension.

Now we will make four figures from the uou platform.

uou left, left index pushed down and up through, ch right

Note the knot is a granny knot and the heart can slide in either direction. The center heart is stable under tension.

uou left, left index pushed down and up through, ch left

Note the knot is a square knot and the heart cannot slide out in either direction. The center heart suffers intensionality.

uou right, right index pushed down and up through, ch right

Note the knot is a granny knot and the heart cannot slide in either direction. The heart suffers intensionality.

uou right, right index pushed down and up through, ch left

Note the knot is a square knot and the heart can slide in either direction. The figure is stable under tension.

In summary the following table indicates salient differences.

The pattern is a complex one and you should consider it carefully.

If the beginning move is done with the opposite hand to the chopstick heart ending move the heart is stable and free. When they are done with the same hands the hearts are unstable and caught.

The knots which result show no such pattern.

i began with these eight figures for this paper because they represent two of my passions. i have been intrigued with forming heart figures for quite some time now, and for a long time i tried to form a granny knot within a string figure. Although it was proved to me that forming a granny knot from a null knot (circle or torus) of string was impossible i am a stubborn man and went to the trefoil knot to find a way to have granny knots in my figures. The hearts in the above figures were simply icing on my cake.

Now i would like to explore further the possibilities of making figures i think interesting from the trefoil loop of string.

What if we merely proceeded from the original four opening positions without the pushing of the index loops down through the other loop. What would we get from the chopstick heart moves on the four beginning positions.

ouo left, ch right

Note square knot, and hoop loops are free to pass to the right both going under the right side of the knot, but are caught moving to the left. The top left string of the left hook goes over the left side of the knot and under then over at the hook.

But now when we use the chopstick heart left finishing move a strange thing happens.

ouo left, ch left

Now the left side of the figure collapses and the knot is a granny with only the right side of the knot held, the top string going over then under the right side of the knot.

ouo right, ch right

Here we have the right side collapse, the knot is a square knot, and the left top string goes over then under the left side of the knot.

ouo right, ch left

This is a granny knot and the left hooked loop is free, both strings passing under the left side of the knot. The right top string of the right hooked loop passes over then under the right side of the knot.

I will leave it to the reader to form the other four figures of the octet and will show only my summative table of results for all eight.

Now compare this to the table of our first eight figures.

The same pattern of knots occurs for the pairings of beginning moves and chopstick heart moves and the stable hearts and freely moving hearts in the first eight figures are the same as the center hook in the last eight figures.

Now there has begun to emerge similarities within the patterns when an exhaustive survey of all the variants in a particular set of figures is made, let’s explore more briefly other ideas.

ouo left, both index – 1/2, ch right

The center portion where the two opposing hearts is of interest. This and the following few pictures can be distinguished by the difference in the crossing patterns in this area.

ouo left, both index – 1/2, ch left

Note the top right string entering the center now has a slightly different path.

ouo right, both index – 1/2, ch right

Note the top left string entering the center now has a slightly different path.

ouo right, both index – 1/2, ch left

Note center four star is almost completely interwoven. i couldn’t find a method for having this star completely interwoven with my trefoil loom. i conjecture it is impossible in a trefoil loom environment just as it is impossible to form a granny knot in the null knot (usual string figure) environment. i leave it to my readers to explore the other similar figures formed with both index rotated minus one half.

Now let’s try another variant.

ouo left, both index + 1/2, ch right

A different yet interesting figure with a single heart and a loop.

ouo left, both index + 1/2, ch left

Now we have two separate hearts offset on either side of the figure. So now we can predict that with the ouo right beginning the ch right ending will produce two hearts and the ch left ending will have only one. I will show only the two heart figure.

ouo right, both index + 1/2, ch right

You would expect the ch left variant of the figure immediately above to have only one heart and you would be correct.

But now let’s move on to another set of figures with a different index rotation.

ouo left, both index minus 2/2, ch right

Now we have a square knot squeezed in the center by the two hooking loops. Note the loops would hook if there were no knot strings to impede them.

ouo left, both index – 2/2, ch left

Now you have a granny knot squeezed and the two squeezing loops would not hook if the knot did not impede their movement.

ouo right, both index – 2/2, ch right

Now we have a square knot again, and if you look carefully the two squeezing loops would not hook if unimpeded in their movement.

And by now you should expect the following result.

ouo right, both index – 2/2, ch left

Yes, it is a granny knot with the two sqeeezing loops hooking if unimpeded.

So now let’s play around with a fresh conjecture, rotating only one index loop.

uou left, left index only + 2/2, ch right

Only the left side of the square knot is caught and the two center loops are hooked if unimpeded.

uou left, left index on.y + 2/2, ch left

Only the left side of this granny knot is held and the two center loops would not hook if unimpeded.

More Complex Knot Looms

Here i will leave my explorations of figures formed from the trefoil knot loom. There are many, many more ways to make figures with this loom but i have shown a few of the “low hanging fruit” possibilities and will now segue into more complex loop positions.

The trefoil knot is the simplest knot which cannot be transformed into the null knot. There are also composite knots. The simplest of these are the granny knot and the square knot. They are both made up of two trefoils introduced into the same loop of string. The following are the four varieties of the simplest trefoil knot :

granny uou uou square uou ouo

The left knot is a granny knot and i characterize it as uou uou with the first wrap on top of the figure read using the bottom left string entering the wrap from left to right (under then over then under), the second wrap is the bottom one read similarly.

The right knot is a square knot which i characterize as uou ouo. When i began making figures with this loom i placed the thumbs down into the top circle, the index fingers down into the middle circle and the little fingers down into the bottom circle.

There are two other looms i have not shown here. They are granny ouo ouo and square ouo uou. Each loom has to be made by forming the wraps before the ends of the string are fused.

Now i show some of the more obvious figures to me made from these looms.

square uou ouo, ch right

Remember after fixing the bottom both index strings have to be rotated – 1/2 to normalize the loom before the figure is formed.

Note it is a square knot that is formed sand the figure is stable. the inner heart cannot move left but is free to move right with both its forming strings passing under the right side of the heart.

Then the following seven figures finish the octet of difference figures able to be formed using the four looms and the ch right and ch left.

square uou ouo, ch left

This is a granny knot and the inner heart has failed under intensionality with the right loop unable to move freely to the right.

square ouo uou, ch left

This is a granny knot with the inner heart failing under intensionality with the left loop unable to move freely to the left.

square ouo uou, ch left

This is a square knot and the figure is stable. The right side of the heart cannot move to the right but is free to move to the left.

granny ouo ouo, ch right

This is a granny knot which is stable and the heart can slide to the right but not to the left.

granny ouo ouo, ch left

This is a square knot and the heart is distorted by intensionality. The left hook would slide free and the right would catch on the side of the knot.

granny uou uou, ch right

This is a square knot and the inner heart yields to intensionality. The right hooking string would go free while the left would catch on the left side of the knot.

granny uou uou, ch left

This is a granny knot which is stable. The heart is free to move left but not to the right.

So now let’s summarize the differences of these last eight figures in a table.

This result is fairly straightforward considering what we have been discovering in the earlier figures. Nothing here surprised me.

Now for one more octet to compare results from. i will illustrate only the first two and adduce a table for the usual array of results for all eight. The reader may wish to experiment to find this information out for himself or herself.

square uou ouo, right index down through right little finger, ch right

This is a square knot with the center loops in a single wrap and caught on both sides.

square uou ouo, right index down through right little finger, ch left

This is a granny knot and the usual center loops have not caught and the sides of the figure do not have the characteristic lower double wrap and upper single wrap necessary for its form to blossom.

As you can see the results follow the same pattern.

So now let’s try some other variations.

granny-ouo-ouo, right index down through right little finger to simplify, left index – 1/2, right index + 1/2, ch right

In this figure i was finally able to make a navajo four star which is completely and tightly interwoven. The previous attempts with the simple trefoil loom were not. I was ecstatic at fingding this figure since i had been trying to find such a figure for the last fifteen years or so, much like i had been hunting for a figure with a square knot. Now that i have begun to explore the geometries possible with knotted looms i have finally scratched these itches.

A brief exploration of the other similar figures from the composite knot looms did not yield another example of a tightly woven navajo four square. i conjecture there are others but leave it to the readers (and to myself) to find them with more thorough searches.

And for further investigations into these composite knot looms i have tentatively outlined for myself the following loom positions (which i feel promise to be more difficult to manipulate into pleasing simple geometries).

These five looms are all the square ouo uou loom ready for thumbs in top loop, index fingers in middle loops and little fingers in bottom loop.

And now i end this paper. i feel i have given a sense of how knotted loops of string can be used to form hitherto impossible figures to form. The search for these figures led me to this unorthodox approach in order to find them. i hope i will not be thrown out of the string figure coterie for this transgression.