- mathematics of the unknot under tension

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

The following paragraphs are offered as a brief introduction to - or as a concise summary of - the full version of this paper. With the exception only of the ethnographical figures and the use made of knot theory, the contents are due solely to the present author.

Many **ethnographical string figures** have come down to us only in the form of an ambiguous photograph in which, at the crossings, it is impossible to determine by eye which string lies over which. A knowledge of the set of all possible similar-looking string figures is a considerable aid in attempting to reconstruct such a figure. The earlier parts of this paper shows how such a set (the set of all possible look-alikes) may be determined mathematically.

We begin by defining a **string figure **as an *unknot under tension*: that is, as a design made from an unknotted loop of string and held, or otherwise supported, in a state of tension.

String figures may be two or three dimensional. In three dimensional figures the points of support will not all lie in a plane.

Example. Stone Money (fig. 1) is a two dimensional figure: the points of support (not labelled) are at the four 'corners'.

The constituent parts of a string figure are the **motifs** and the **3D framework**.

Example. The motifs of Stone Money (fig. 1) are labelled A to H. The figure is two dimensional, so there is no 3D framework.

A **motif **is a substructure which is

- two dimensional (ignoring minor irregularities due to the thickness of the string)
- connected to the remainder of the string figure by free single segments of string
- reflectible in the sense that the silhouette of a string figure in a plane parallel to the sub-structure is unaltered by the instantaneous replacement of the substructure by its reflection (the reflection taking place in the plane of the substructure)
- a minimal such structure.

Example. In Stone Money (fig. 1), the motif F (a double crossing) is joined to the remainder of the figure by the four free single segments running from F to the motif E, to the motif H, to the motif D, and to the lower right point of support. The motif F and the four supporting segments constitute a two-dimensional sub-structure. The silhouette of the figure is unaltered if motif F is replaced by motif f (see fig. 2 for an example of f): the figure remains stable (though not necessarily constructible from an unknotted loop of string) and no change of silhouette will occur. F is a minimal such structure since the replacement of either one crossing of F by its reflection would cause the figure to collapse, and thus a change of silhouette will occur.

The **3D framework **of a string figure is made up of all those contacts in a three-dimensional string figure in which the single segments of string diverging from a contact do not all lie in a single plane.

The **string contacts **that occur in a string figure are resolved into **string crossings **in two circumstances:

- when considering the motifs (which are two dimensional)
- when considering the projection of a string figure onto two dimensions.

**Look-alikes **are defined as string figures that differ only in the parity of one or more motifs.

Example. The look-alikes Stone Money and Star (figs 1 and 2) differ only in the parity of motifs E, F, G and H.

The **set of all possible look-alikes **of a given string figure can be determined by using the techniques and tools of knot theory. We start by labelling the motifs, then project the string figure onto two dimensions, ensuring no two projected string crossings lie upon one another, and noting which of the projected crossings originates from each motif. Every possibility for changing the parity of one or more motifs is then investigated. The knottedness of each outcome is tested (using the tools of knot theory): if the result is an unknot, the combination of motifs changed corresponds to a look-alike of the original string figure.

Example. In the case of Stone Money, a two dimensional figure, the projection onto two dimensions will be exactly the sketch above (fig. 1). The crossings are grouped into eight motifs. One of the possibilities to be examined will be the changing of the crossings in motifs E, F, G and H: changing the parity of these four motifs results in an unknot, showing that fig. 2 is a constructible string figure, and a look-alike of Stone Money.

Having shown how the set of all look-alikes may be determined **mathematically**, the paper moves on to show how look-alikes of a given figure may be obtained **manipulatively**.

The **Deconstruction Theorem **(introduced in Part II) states that, if a subset of motifs may be unravelled without interfering with the remainder of the string figure, then that subset may be reformed such that each motif in the subset is in a state of opposite parity.

Example. In Stone Money (fig. 1) ABCD may be unravelled, then reformed in a state of opposite parity, to give the look-alike abcdEFGH (see Part IV, Example 2, for details).

This interesting theorem gives rise to some additional problems:

- A mathematical, as opposed to a mechanical, determination of
**the possible partial unravellings**of a given string figure remains to be established. - The
**conjecture**is advanced that repeated applications of the Deconstruction Theorem enable any one of a set of look-alikes to be transformed into any other look-alike of the set.

Detailed explanations, proofs, additional results, and numerous examples, can be found in the full version of this paper.

String Figure Mathematics - full version - Part I

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