Example 1. Fig. 23 is obtainable from the Brown Bear (fig. 7) by unravelling motifs D, E and F. The set of all lookalikes of fig. 23 is {((A)BC)} È {(AB((C)} = {ABC, ABc, Abc, aBC, abC, abc}. The sequences that relate the six lookalikes of fig. 23 are shown on the right below. For example, the lookalike Abc can be transformed into the lookalike 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 lookalikes 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 singlecrossings (G, H), four doublecrossings (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 lookalikes, 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 lookalikes have been identified by this process: the remaining 232 (2^{8}  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 lookalike ABCDefgh was recorded from Guyana by Roth and published in 1924 (see our fig. 25; Roth's fig. 267 is incorrect).
Fig. 24  Stone Money

Fig. 25  Star

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 lookalike abC, then unravel C, then reform C in the reflected state, then reform AB in the unreflected state; the result is the lookalike abc.
Using the Brown Bear as an example, we have such relationships between the lookalikes (derived from the unravelling process) as:
Further relationships may be algebraically deduced from these. For example:
If conjecture 3 above is correct, relationships may be similarly derived between any arbitrary pair of lookalikes of a given string figure.
We have shown how knot theory may be used to determine the set of lookalikes of a given string figure. In addition we have proved and demonstrated that one lookalike 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.
The author has also written extensively on juggling and the mathematics of juggling in Four Ball Juggling, From Simple Patterns To Advanced Theory (700 illustrations, 202 pages, ISBN 0952486008), available from the second hand market.