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

Opening A

 - - - - - .. - - - - - -
'                        '
'                        '
'    AB - - A. - - Ab    '
'    '      '      '     '
'    '      '      '     '
 - - .B     ..     .b - -
     '      '      '
     '      '      '
     aB - - a. - - ab

Jayne’s Tipstaff

                        aBC
                         '
                         '
ABC - - A.. - - ... - - a.. - - abc
         '
         '
        Abc

Jayne’s No Name

           ABCDEF                abcDEF
             '                     '
             '                     '
ABCDef - - ABCD..                abcD.. - - abcDef
             '                     '
             '                     '
           ABC... - - ...... - - abc...
             '                     '
             '                     '
ABCdEF - - ABCd..                abcd.. - - abcdEF
             '                     '
             '                     '
           ABCdef                abcdef

Moon Rising over Whale Carcass

• The illustration (fig. 18) shows Moon Rising over Whale Carcass (an Alaskan string figure collected in 1913-14 by Diamond Jenness and published in 1924. Jenness' fig. 89 is incorrect).
• The set of all look-alikes = {(((AB)CD)EF)} È {((A(BC)D)EF)}.
• There are two shards, ABCD and EF, with catalogues {ABCD, ABcd, AbcD, aBCd, abCD, abcd} and {EF, ef}.
• The number of impossible string figures masquerading as look-alikes of Moon Rising over Whale Carcass is 52 (2⁶ - 12), one of them being the impossible figure ABCdEF as illustrated by Jenness.

Jayne’s Worm

This figure, Jayne’s (Second) Worm, is illustrated again to demonstrate the application of the analysis to a three-dimensional figure.

• The 3D frameset P₁P₂P₃P₄P₅P₆ can be unravelled to leave ABC, then BC can be unravelled to leave A: hence {[((A)BC) P₁P₂P₃P₄P₅P₆]} is a subset of (four) look-alikes, the square brackets indicating a non-reflectible substructure. There are no other look-alikes.
• There are three shards, A, BC and P₁P₂P₃P₄P₅P₆, with catalogues {A, a}, {BC, bc} and {P₁P₂P₃P₄P₅P₆}.

Hermaphrodites

String figures can be constructed with as many shards as desired. Hermaphrodites (collected in Hawaii by Lyle Dickey in 1915-17) is an asymmetric figure based upon a repetitive weaving process capable of endless repetition. The weaving process generates an additional two shards at each iteration.

• Hermaphrodites has thirteen motifs: seven double-crossings (A, C, E, G, I, K and M) and six triple-crossings (B, D, F, H, J and L).
• AB may be unravelled, then CD, then EF, then GH, then IJ, then KL, then M.
• There are six shards: AB, CD, EF, GH, IJ and KLM. The corresponding catalogues are {AB, ab}, {CD, cd}, {EF, ef}, {GH, gh} and {IJ, ij}, all of order 2, and {KLM, KLm, KlM, kLm, klM, klm} of order 6.
• The weaving process, if continued, adds further shards with catalogues of order 2 at the left of the figure.

Jayne’s Pygmy Diamonds

Pygmy Diamonds has symetrically placed motifs. The look-alikes of such a 'symmetric' string figure may be grouped by rotational equivalence, one look-alike of a group being transformable into another by rotating the figure in space.

• The set of all look-alikes = {(((A(B(C)))DE)FG)} È {(AB(Cd(((E))F(G))))}. Eight look-alikes belong to both subsets, those in the subset {(AB(Cde(FG)))}. There are thus 56 (32 + 32 - 8) distinct look-alikes: that is, there are 56 ways in which the parities of the seven motifs may be changed to generate a look-alike.
• Practical tip: To unravel FG from ABCDEFG, hold crossing E tightly as if the strings are glued, then twist E through 360 degrees until F and G come undone. (A construction of Pygmy Diamonds in given in the next section.)
• The 56 look-alikes of Pygmy Diamonds may be grouped by rotational equivalence. The set {((A(B(C)))DEFG)} of sixteen (2⁴) look-alikes contains one look-alike from each such group.

Look-alike generator for Jayne’s Pygmy Diamonds

Each of the sixteen rotationable look-alikes may be constructed by adopting none, one or more of the alternatives at steps 1, 2, 3 and 4 in the ‘look-alike generator’ below. Step 1 affects the parity of motif A; step 2 affects that of motif B; step 3 affects that of motif C; while step 4 affects the parity of the set of motifs DEFG. The digits in the binary representations (fig. 22) refer, from right to left, to steps 1, 2, 3 and 4 in the look-alike generating procedure below: a 0 indicates that the first movement in the step is to be used, a 1 that the alternative movement [given in square brackets] must be adopted. Thus 0100 indicates the adoption of the alternative at step 3.

String figure glossary

1. Insert little fingers into the loop. Insert the left [right] thumb from above into the right [left] little finger loop, then hook up the near right [left] little finger string by rotating the left [right] thumb towards you and up. Insert the right [left] thumb from below into the left [right] thumb loop. Extend.
2. Give the right thumb loop a full turn (360 degrees) away from [towards] you.
3. Give the right little finger loop a full turn (360 degrees) towards [away from] you.
4. Pass the right little finger loop up [down] through the right thumb loop and return to the right little finger, then pass the right thumb loop up [down] through the right little finger loop and return to the right thumb.
5. With each index remove the thumb loop from above. Pass each thumb under all intervening strings and from below up into the little finger loop, and then from below up into the index loop. Release the index fingers. Hold each thumb against the base of the index finger to prevent the far thumb strings from slipping during the next action. On each side one far thumb string runs towards the centre of the figure. Pass each index down on the far side of that far thumb string and pick up the string on the tip of the index. Raise the upper frame string high on the tips of the index fingers.

0000	0001	0010	0011
0100	0101	0110	0111

Fig. 22 - Eight of the sixteen rotational look-alikes of Pygmy Diamonds