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

by Martin Probert

Part V
Conjectures concerning the unravelling process

  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(li) is the operation of unravelling successive subsets of motifs from a look-alike li 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.

string figure
Fig. 23

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).

string figure

string figure

Fig. 24 - Stone Money
Caroline Islands 1902

Fig. 25 - Star
Guyana 1907-24

A functional notation for the unravelling process

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:

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 look-alikes of a given string figure.

Conclusions

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.


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 0-9524860-0-8), available from the second hand market.

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