String Figures and Knot Theory
 mathematics of the unknot under tension
by Martin Probert
Part III
String Figures Analysed
Opening A
Fig. 15  Opening A and lookalikes
Jayne’s Tipstaff
Fig. 16  Tipstaff or Truncheon
Great Britain c. 1860
 The illustration (fig. 16) shows Jayne’s Tipstaff. The original is, however, considerably elongated compared with our illustration.
 The set of lookalikes of the Tipstaff
= {((A)BC)}
= {ABC, Abc, aBC, abc}.
 There are two shards, A and BC,
with catalogues {A, a} and {BC, bc}.
 The sequences of 'unravellings by motifs' that relate the four lookalikes are given on the right.

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

Jayne’s No Name
Fig. 17  Jayne's No Name (one half unravelled)
Caroline Islands 1902
 The set of lookalikes of fig. 17
= {(((ABC)D)EF)}
= {ABCDEF, ABCDef, ABCdEF, ABCdef, abcDEF, abcDef, abcdEF, abcdef}.
 There are three shards, ABC, D and EF,
with catalogues {ABC, abc}, {D, d} and {EF, ef}.
 The sequences of 'unravellings by motifs' that relate the eight lookalikes are given on the right.

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

Moon Rising over Whale Carcass
Fig. 18  Moon Rising over Whale Carcass
Alaska 191314
 The illustration (fig. 18) shows Moon Rising over Whale Carcass (an Alaskan string figure collected in 191314 by Diamond Jenness and published in 1924. Jenness' fig. 89 is incorrect).
 The set of all lookalikes = {(((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 lookalikes of Moon Rising over Whale Carcass is 52 (2^{6}  12), one of them being the impossible figure ABCdEF as illustrated by Jenness.
Jayne’s Worm
Fig. 19  A Worm
North America 1904
This figure, Jayne’s (Second) Worm, is illustrated again to demonstrate the application of the analysis to a threedimensional figure.
 The 3D frameset P_{1}P_{2}P_{3}P_{4}P_{5}P_{6} can be unravelled to leave ABC, then BC can be unravelled to leave A: hence {[((A)BC) P_{1}P_{2}P_{3}P_{4}P_{5}P_{6}]} is a subset of (four) lookalikes, the square brackets indicating a nonreflectible substructure. There are no other lookalikes.
 There are three shards, A, BC and P_{1}P_{2}P_{3}P_{4}P_{5}P_{6}, with catalogues {A, a}, {BC, bc} and {P_{1}P_{2}P_{3}P_{4}P_{5}P_{6}}.
Hermaphrodites
Fig. 20  Hermaphrodites
Hawaii 191517
String figures can be constructed with as many shards as desired. Hermaphrodites (collected in Hawaii by Lyle Dickey in 191517) 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 doublecrossings (A, C, E, G, I, K and M) and six triplecrossings (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
Fig. 21  Pygmy Diamonds
Congo Kasai Valley 1904
Pygmy Diamonds has symetrically placed motifs. The lookalikes of such a 'symmetric' string figure may be grouped by rotational equivalence, one lookalike of a group being transformable into another by rotating the figure in space.
 The set of all lookalikes = {(((A(B(C)))DE)FG)} È
{(AB(Cd(((E))F(G))))}. Eight lookalikes belong to both subsets, those in the subset {(AB(Cde(FG)))}. There are thus 56 (32 + 32  8) distinct lookalikes: that is, there are 56 ways in which the parities of the seven motifs may be changed to generate a lookalike.
 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.
 The 56 lookalikes of Pygmy Diamonds may be grouped by rotational equivalence. The set {((A(B(C)))DEFG)} of sixteen (2^{4}) lookalikes contains one lookalike from each such group.
Lookalike generator for Jayne’s Pygmy Diamonds
Each of the sixteen rotationable lookalikes may be constructed by adopting none, one or more of the alternatives at steps 1, 2, 3 and 4 in the ‘lookalike 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 lookalike 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
 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.
 Give the right thumb loop a full turn (360 degrees) away from [towards] you.
 Give the right little finger loop a full turn (360 degrees) towards [away from] you.
 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.
 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 lookalikes of Pygmy Diamonds
String Figure Mathematics  Part IV
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