String Figures and Knot Theory
- mathematics of the unknot under tension

In 1920 the historian of mathematics W. W. Rouse Ball wrote of string figures that "What ... is striking, is the immense variety of well-defined patterns". Yet no mathematical result that applies equally to all string figures (two dimensional and three dimensional, symmetric and asymmetric) appears ever to have been published. This paper attempts to rectify the situation. With the exception only of the ethnographical figures and the use made in one paragraph of knot theory, the contents are due solely to the present author.

Part I defines string figures and introduces the concepts of look-alikes and motifs. Part II shows how look-alikes may be determined by knot theory and introduces a process by which one look-alike is converted into another. Part III uses the results derived to analyse a range of string figures. Part IV contains several conjectures concerning the partial unravelling of string figures.

From the viewpoint of knot theory every string figure - each one of the 2000 known figures and every figure yet to be invented - is the ‘trivial knot’, an unknotted loop of string. To differentiate between one string figure and another we shall take account of the effect of tension as the string figure is extended between a number of suspension points.

Definitions:

1. A string figure is a pattern formed by ‘weaving on the hands a single loop of string’; in the finished figure the strings show ‘the effects of strain and of deflection produced by crosses, knots and twists’ (Caroline Furness Jayne, String Figures, 1906. Dover reprint ISBN 0-486-20152-X, 1962).
2. A string figure is an unknot under tension (Martin Probert, 2002).

We emphasise the following characteristics of string figures:

1. A string figure is a structure displayed under tension. Limp designs, such as those formed by arranging a loop of string on a horizontal surface, while examples of trivial knots, are not here considered as examples of string figures.
2. A string figure is a supported structure (usually on the hands). The points of suspension are an intrinsic part of the figure: in most cases a string figure that is removed from the hands, turned over, and replaced, produces a different figure needing a different method of construction. This is quite different from the knots of knot theory. It is a mistake to confuse the counting of string figures with the counting of knots. For example, Opening A and the variation in which the left index performs its action before the right index are distinct string figures (illustrated later in Part III fig 15 left and right). For a consideration of rotation in the context of string figures, see the two sections in Part III on Jayne's Pygmy Diamonds.

It is unlikely that it is possible to produce a unique and simple mathematical description of a string figure. A string figure is an unknot under tension. Any description of the static final figure would need to take account of string contacts, points of support, tensions in the string segments, and the three dimensional nature of many figures. The tensions in particular are very subtle: a small change in tension can produce a significant change in the figure. A notation defining the two dimensional projection of a three dimensional figure would give insufficient information to allow the figure to be constructed. And a description based upon a method of construction would certainly not be unique, for there are many ways in which a string figure may be constructed.

‘Spot the difference’ is a familiar game from childhood: the challenge is to spot the difference in two similar pictures. A similar game may be played with string figures. What is the difference in the following two string figures?

Fig. 1 - Stone Money Caroline Islands 1902	Fig. 2 - Star Guyana c.1924

The labelling indicates the points at which the figures differ. E, F, G and H (fig. 1) differ from e, f, g and h (fig. 2). Such similar looking figures will be called look-alikes. We shall study such figures in detail.

The substructure efgh (fig. 2) is the reflection of EFGH (fig. 1), the reflection taking place in the plane of the substructure.

We define a reflectible substructure as one with the following three characteristics:

1. A reflectible substructure is two-dimensional (allowance being made for inconsequential irregularities due to the thickness of the string). The string figure of which the substructure is part may however be three-dimensional.
2. A reflectible substructure is connected to the remainder of the string figure by free single segments of string. These free single segments lie in the same two-dimensional plane as the substructure supported by them.
3. The sillouette of a string figure in a plane parallel to a substructure is unaltered by the replacement of the substructure by one of opposite parity.

A pair of reflectible substructures such as EFGH and efgh will be called substructures of opposite parity.

1. Any one or two of the three string contacts occuring at D (fig. 1) constitute a non-reflectible substructure: attempting to replace such a substructure by its relection causes a change of 'sillouette' (i.e. the string figure collapses).
2. A three-dimensional substructure is also non-reflectible since the free single segments connecting the substructure to the remainder of the string figure would no longer connect after reflection.

An examination of Stone Money (fig. 1) shows that A, B, C, D, E, F and G are all reflectible substructures. There are no smaller reflectible substructures than those labelled A to G. Such minimal reflectible substructures will be called motifs.

It is easily proved that the reflectible substructures of any string figure are partitioned into motifs. The proof, which follows from the definitions of reflectible substructures and motifs, is left to the reader.

If a motif is denoted by an upper case letter, then the substitution in a string figure of that motif by one of opposite parity will be denoted by a lower case letter (compare fig. 1 motif G with fig. 2 motif g). A set of motifs will be denoted by a succession of upper or lower case letters where each letter is the label of a motif (e.g. fig. 2 contains the set of motifs ABCDefgh).

Motifs occur at each point of support of a string figure but, in such a case as Stone Money (fig. 1) where a single string encircles each point of support, these motifs are trivial and have not been labelled.

A motif may contain a complex cluster of string contacts. Two or more string segments may, for example, twist and intertwine ropewise around one another. We shall only represent such clusters in terms of distinct string crossings when we discuss the projection of a figure onto two dimensions for analysis by knot theory. To justify certain processes involving the unravelling of a three dimensional string figure, we shall make use of the Deconstruction Theorem introduced in Part II.

Four motifs we shall meet frequently are illustrated in fig. 3. From left to right the four motifs will be referred to as a single-crossing, a double-crossing, a spiral triple-crossing and a triangular triple-crossing.

Infinitely many different motifs exist. For example, two segments of string might twist around one another in spiral fashion through any multiple of 180 degrees to create a motif.

Each motif of fig. 3 above together with the corresponding motif of fig. 4 below make up a pair of motifs of opposite parity.

String figure glossary

• In the Worm (collected by CF Jayne, but Jayne's fig. 689 is incorrect) there are six points of support (where the string loops around 1, 2 and 5).
• There are three motifs: a single-crossing at A, a triple-crossing at B and a double-crossing at C.
• The substructures in the locality of P₁, P₂, P₃, P₄, P₅ and P₆ are all three-dimensional, and hence non-reflectible.
• Other places in the illustration where strings might appear to be in contact are separated in the three-dimensional figure.

The 3D framework is the set of all those contacts in a three-dimensional string figure in which the single segments of string diverging from the contact do not all lie in a single plane. The 3D framework in the Worm is made up of the contacts P₁, P₂, P₃, P₄, P₅ and P₆.

Motifs in corresponding positions in two look-alikes are equivalent if, without involving the supporting single strands, the interiors of the motifs may be manipulated into identical configurations. The motifs in fig. 6 are equivalent.

We are not here concerned with the identification of the motifs in a given string figure. Identification of the motifs is merely a process of establishing the minimal substructures with the reflectible properties listed earlier. We assume that the motifs have been identified and pursue the following questions:

Changing the parity of one or more motifs will result in one of two outcomes: a similar looking string figure, or an impossible figure (i.e. one that may not be constructed from an unknotted loop). How may the similar looking string figures be identified, and what relationships exist between them?