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

1. If no non-empty proper subset of motifs of a string figure F is unravellable, then the only other possible look-alike of F is the reflection of F. (A distinct look-alike of opposite parity will only exist if F is a non-trivial two-dimensional figure.)
2. If the set of all look-alikes of a string figure contains more than two members, then any look-alike in the set contains an unravellable non-empty proper subset of motifs. (The converse is evident from theorem 2 in Part III.)
3. Every look-alike is discoverable through the process of 'unravelling by motifs' described in theorem 2. That is, if U(l_i) is the operation of unravelling successive subsets of motifs from a look-alike l_i and then reforming each two-dimensional subset either in the identical or reflected state (and, in the case of a three-dimensional figure, reforming any three-dimensional subsets of string contacts consisting of entire motifs and/or part or all of the 3D framework of the string figure in their original states), then for any two look-alikes l₁ and l₂ there exists a series of n such operations such that l₂ = Uⁿ (l₁).

Example 1. Fig. 23 is obtainable from the Brown Bear (fig. 7) by unravelling motifs D, E and F. The set of all look-alikes of fig. 23 is {((A)BC)} È  {(AB((C)} = {ABC, ABc, Abc, aBC, abC, abc}. The sequences that relate the six look-alikes of fig. 23 are shown on the right below. For example, the look-alike Abc can be transformed into the look-alike ABc as follows: from Abc unravel bc and form BC to give ABC, then from ABC unravel AB then C and form c then AB to give ABc. If conjecture 3 is correct a similar sequence of operations may be found to relate an arbitrary pair of look-alikes of any string figure.

abC - - ..C - - ... - - ..c - - ABc
         '               '
         '               '
        ABC             abc
         '               '
         '               '
Abc - - A.. - - ... - - a.. - - aBC

Example 2. Jayne’s Stone Money (see our fig. 24; Jayne's fig. 359 is incorrect) has eight motifs: two single-crossings (G, H), four double-crossings (A, B, E, F) and two triple crossings (C, D). (For a construction that results in a string figure identical to Stone Money, see the 'plinthios' on this web site.) In Stone Money ABCD may be unravelled (a good grip at G and a twist at G will release the final crossings from C and D), then H, then EF, then G. Alternatively EFCD may be unravelled, then G, then AB, then H. The set of all look-alikes, if conjecture 3 is correct and no possibility of unravelling has been missed, equals {(ABCD((EF(G))H))} È {(((AB))CDEF(G(H)))}. 24 distinct look-alikes have been identified by this process: the remaining 232 (2⁸ - 24) variants would then all be impossible string figures. The shards are then ABCDEF, G and H with catalogues {ABCDEF, ABCDef, abcdEF, ABcdef, abCDEF, abcdef}, {G, g} and {H, h}. The look-alike ABCDefgh was recorded from Guyana by Roth and published in 1924 (see our fig. 25; Roth's fig. 267 is incorrect).

We briefly outline a functional notation for the process of unravelling by motifs.

Let X: represent the unravelling of substructure X, X^-1: the reforming of the substructure in the reflected state, and X^-1: the reforming of the substructure in the unreflected state. Thus AB^-1: C^-1: C: AB: (abC) means first unravel ab from the look-alike abC, then unravel C, then reform C in the reflected state, then reform AB in the unreflected state; the result is the look-alike abc.

Using the Brown Bear as an example, we have such relationships between the look-alikes (derived from the unravelling process) as:

• AbcDef = EF^-1: D^-1: BC^-1: BC: D: EF: (ABCDEF)
• abCDEF = EF^-1: D^-1: AB^-1: AB: D: EF: (ABCDEF)

Further relationships may be algebraically deduced from these. For example:

• abCDEF = EF^-1: D^-1: AB^-1: AB: BC^-1: BC: D: EF: (AbcDef)

If conjecture 3 above is correct, relationships may be similarly derived between any arbitrary pair of look-alikes of a given string figure.

We have shown how knot theory may be used to determine the set of look-alikes of a given string figure. In addition we have proved and demonstrated that one look-alike may be transformed into another by the process of unravelling by motifs. What remains to be established is a mathematical, as opposed to a mechanical, determination of the possible partial unravellings of a given string figure.