WITH A CENSUS OF THE AMPHICHEIRALS WITH TWELVE CROSSINGS. 253 
§ 7. Census of Amphicheirals with Twelve Crossings. 
In the census of the amphicheiral knots of the first order with twelve crossings 
the alternate crossings which occur as the knot is described in a fixed direction from 
the amphicheiral centre at infinity are denoted by a, b, c, . . . The first crossing a 
in the amphicheirals of the second order has been assigned arbitrarily. Only the 
letters which occupy the even places in the sequence of the alphabetical symbol of 
the knot are given, although the distortions D n , given in brackets, are detected only 
in the complete scheme. The number of different forms of a given knot is indicated 
by the number of distinct distortions D n in the brackets following the knot scheme ; 
the different forms are obtained by the product of a distortion D n and its conjugate 
D n , which occurs at the same distance from the amphicheiral centres as D„. 
In the determination of the pairs of amphicheiral centres of a knot the intrinsic 
symbol is very convenient. It may be shown that the sum of the numbers at equal 
distances from an amphicheiral centre is equal to n— 1, where n is the order of 
the knot. From the reduced alphabetical symbols given, it is a simple matter 
to write down the complete alphabetical and therefore the intrinsic symbols of 
the knots ; hence the pairs of amphicheiral centres are known, and the knot may 
be constructed. 
There exist the following amphicheirals with twelve crossings : — 
I. Amphicheirals of the First Order : 
p = 1, Nos. 1-18; p = 2, Nos. 19-54 ; p = 3, Nos. 55-58. 
(1) 
i 
h 
9 l 
3 fc 
c 
b a 
f e 
d 
(2) 
h 
i 
9 k 
l 3 
a 
b c 
d e 
f 
(3) 
i 
9 
h l 
3 k 
c 
a b 
f d 
e 
0) 
h 
i 
9 k 
1 3 
b 
a c 
df 
e 
PI 
d 2 *i 
>5 
0) 
i 
9 
h l 
jk 
c 
b a 
f e 
d 
Pll 
(6) 
h 
a 
9 p 
l 3 
d 
bf 
i e 
k 
P$ 
(7) 
d 
h 
g a 
k j 
b 
c l 
e f 
i 
(8) 
d 
h 
& 9 
k i 
a 
c l 
« 3 
f 
p>:. 
1 ’ 
K 
» 
(9) 
9 
h 
b a 
k l 
d 
o f 
e 3 
i 
pt 
dJ. 
, i)“, r 
(10) 
c 
h 
a g 
k i 
d 
bf 
l e 
k 
tK 
* 
, uX, 
■ DM 
(11) 
i 
h 
f l 
k c 
3 
b a 
9 « 
d 
pj 
(12) 
c 
h 
* 3 
k d 
a 
b 9 
l d 
f 
p>:. 
K 
K 
(13) 
e 
a 
9 b 
c 3 
d 
U 
h i 
k 
[K 
K 
d;j 
(14) 
e 
d 
9 a 
b 3 
c 
l h 
f h 
i 
K 
Df] 
(15) 
e 
9 
b a 
d k 
c 
l h 
f 3 
i 
[Df, 
Df 
J3 b , 
, dX] 
