- mathematics of the unknot under tension

First posted February 2001. Last revised December 2002.

## AbstractMany
The contents are due solely to the present author. It is believed that these are the first mathematical results published that apply equally to all string figures (two dimensional and three dimensional, symmetric and asymmetric). It is hoped that these results will encourage others to develop the subject. |

Identifying String Figure Look-Alikes Mathematically

"and Alice knew which was which in a moment"

(Lewis Carroll: Through the Looking-Glass, Chapter IV, Tweedledum and Tweedledee)

What is the difference in the two figures illustrated below?

Fig. 1 - Stone Money
Caroline Islands 1902 (Jayne) |
Fig. 2 - Star
Guyana c.1924 (Roth) |

The labelling indicates the points at which the figures differ. E, F, G and H of Stone Money (see our fig. 1; Jayne's fig. 359 is incorrect) differ from e, f, g and h of Star (see our fig. 2; Roth's fig. 267 is incorrect). Such similar looking figures will be called **look-alikes**.

The question addressed in this paper is: Does such a string figure look-alike as ABCDeFGH exist? (That is, a string figure similar to fig. 1 but in which E is replaced by e of fig. 2.)

The only previous way to determine the answer to such a question has been to weave the figure from a length of string, tie the ends once the figure has been formed, then attempt to unravel the figure to see if a single loop can be obtained. In this paper I introduce two new procedures: (i) using the techniques of knot theory to determine all possible look-alikes, and (ii) weaving new look-alikes from a given figure without the necessity of cutting the loop.

We shall refer to the sub-structures A to H of fig. 1 as motifs. Motifs are determined either by the inspection of existing string figures, or by practical experimentation. The vast majority of known string figures are to a large extent composed of a small set of common motifs. Once the motifs of a figure have been identified, the techniques given in this paper can be applied.We emphasise that the motifs are determined before we begin: only then will we apply the procedures given in this paper. The following sections introduce the concept of a motif. We return to the mathematical determination of the look-alikes in the last section of this part.

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:

- 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).
- A string figure is an
*unknot under tension*(Martin Probert, 2002).

We emphasise the following characteristics of string figures:

**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.**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 IV fig 15 left and right). For a consideration of rotation in the context of string figures, see the two sections in Part IV on Jayne's Pygmy Diamonds.

String figures are loosely described as being either "two dimensional" or "three dimensional". We formalise this by defining a **two dimensional string figure** as one in which the points of support lie on a plane. Due to the thickness of the string, the space occupied by such a figure is not two dimensional: however, if we let the thickness of the string tend to zero, the space occupied by a "two dimensional" figure will tend to two dimensions.

We define a **contact cluster** as a set of string segments in which (i) each segment of the cluster throughout its length is in contact with one or more other segments of the cluster, (ii) every segment of the cluster is in contact either directly or via one or more other segments of the cluster to every other segment of the cluster, and (iii) every segment which can be reached by a series of contacts is part of the cluster. The simplest contact cluster occurs when two string segments cross, making contact at a single point. Stone Money (fig. 1) contains twelve contact clusters: simple clusters at each of G and H, three simple clusters at each of C and D, and a non-simple cluster at each of A, B, E and F.

The substructure efgh (fig. 2) is the reflection of EFGH (fig. 1), the reflection taking place in the plane of the substructure (i.e. in the plane through the points of support). A pair of substructures such as EFGH and efgh will be called **substructures of opposite parity**.

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

- A reflectible substructure consists of a set of
**contact clusters**together with a number of**free single segments of string**that connect the clusters to one another and to the remainder of the figure. - There is a set of points, one on each of the free single segments of string that support the reflectible substructure, that lie on a plane. Thus reflectible substructures are "two dimensional" in the same sense that certain string figures are referred to as "two dimensional" (see above). We stress that the string figure of which the substructure is part may however be three-dimensional.
- The
**silhouette**of a string figure in a plane parallel to a substructure is unaltered by the replacement of the substructure by one of opposite parity.

- Any one or two of the three string contacts occurring at D (fig. 1) constitute a non-reflectible substructure: attempting to replace such a substructure by its reflection causes a change of 'silhouette' (i.e. the string figure collapses).
- 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**.

Motifs, being reflectible substructures, are "two dimensional" (see above).

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 just-formed string figure may need some adjustment to clarify the design. Directions to do so are frequently included among the instructions for forming a figure. Expert string figure makers habitually make the necessary adjustments during the construction process.

What happens if, after such adjustment, the maker remains unclear as to whether a certain reflectible substructure is to be regarded as a single motif, or whether it might be possible to split the reflectible substructure into two or more motifs? * There is no harm in taking the reflectible substructure as a single motif.* The subsequent analysis will simply treat the reflectible substructure as a unit and generate the set of look-alikes in which either that unit or the unit of opposite parity appears.

Each (simple or complex) contact cluster in a string figure is connected to the remainder of the figure by free single segments of string. It may or may not be possible to find a set of points, one on each single segment, that lie on a plane. If such a set does not exist, then the cluster is said to be part of the **3D framework** of the string figure.

Having decided which are the motifs of a given string figure, the set of look-alikes of the figure may now be identified using knot theory:

- Label the motifs.
- Project the string figure onto two dimensions, ensuring no two projected string crossings lie upon one another, and keeping note of which of the projected crossings originated from each motif.
- Investigate every possibility for changing the parity of one or more motifs, and, using the tools and techniques of knot theory, test the knottedness of each outcome: if the outcome is an unknot, the combination of motifs changed corresponds to a look-alike of the original string figure.