WITH A CENSUS OF THE AMPHICHEIRA LS WITH TWELVE CROSSINGS. 241 
while the variants become 
(2') 654566 6 1338543466333 
(3') 4543666335 '6 5 41668333 
(4') 56546661338454366333 
Since the set of numbers (l')is the same as a set of numbers complementary to (4'), 
but in the reverse order, the forms (l) and (4) are equivalent. Likewise (2) and (3) 
represent the same knot. 
Effect of Distortion on the Compartment Symbol . — In a consideration of the 
different forms of a given knot, the compartment symbol is more convenient than the 
alphabetical symbol, although the statement of the effect of a given distortion is now 
not quite so simple. It is clear from figs. 4 and 4' that a reversible tangle R„ exists 
when a compartment is joined once, and only once, to a compartment 8j, which in 
turn is joined at any number of vertices to a compartment <^, and the entire con- 
figuration is bounded on the right and left by the compartments h s respectively, 
\ being joined at the same crossing as S jy let the compartments Sj, \ be 
joined to the reversible tangle R„ by 0, a 2 , a 3 , b 1} b 2 crossings, and to the remainder 
of the knot by af 0, a 3 , 6/, b 2 crossings respectively, so that 
i — rtf + 1 
j = a 2 + 1 
k = a 3 +a 3 ’ 
r = b x + 6/ + 1 
s = b 2 + b 2 +1 
Then the symbol for this portion of the knot becomes 
<V+i 
£ 
f +1 A& j+ &7+i 4 1 
I 
where | indicates that the two compartments may be joined directly or indirectly. 
By the application of the distortion D„, this symbol becomes 
V'+aa 
t 
0 
I 
S a ,4-1 + 
^V+l 
When = a 3 , a 2 = a 3 , b 1 = b 2 , the number of compartments and the number of 
angles in each is undisturbed. If the order in which these compartments are 
joined is the same as for the original knot, then the distortion reduces to a 
deformation, since the two forms are equivalent. 
