WITH A CENSUS OF THE AMPHICHEIRALS WITH TWELVE CROSSINGS. 249 
Of the hundreds of cases arising from the different arrangements of these points, 
and the different ways of joining them, the greater number lead to composite or 
m-filar knots. Also Nos. 19, 21, 22, 24 (p = 2) of the amphicheirals of the first 
order (see Plate) appear among those of the second order, since the arrangement of 
the $ and A. compartments is symmetrical, and hence unaltered by reversion (Tait, 
iii, § 12). Kejection of these reduces the amphicheirals of the second order with 
twelve crossings to the following two : — 
(5) f j gib e l d a c h k [DQ 
(6) fkbiadlehcgj [D*] 
These two knots can be constructed on models involving more than one pair 
of contacts, and hence may be expected to present themselves several times in the 
course of the construction. Starting with a given knot of the second order, the 
different models on which the knot may be constructed are obtained by transferring 
one or more compartments to the inside of the circle, and therefore their corre- 
spondents to the outside of the circle. This amounts merely to a deformation of 
the knot so as to make any desired path into a circle. 
If small letters are used to denote the intersections of the circle and the twin 
pair, while capital letters indicate the points of contact, that is, the crossings at 
which a change of thread takes place, then the different models for the above knots 
may be represented as follows : — 
0) 
A 
f 
b 
j 
c 
G 
l 
h 
d 
i 
(2) 
I 
e 
B 
j 
C 
k 
H 
d 
(3) 
A 
f 
b 
J 
C 
G 
l 
h 
D 
I 
(4) 
A 
F 
E 
B 
j 
c 
G 
L 
K 
H d i 
(5) 
A 
F 
B 
J 
c 
G 
L 
H 
D 
I 
(6) 
A 
F 
E 
B 
J 
C 
G 
L 
K 
H D.I 
(1) 
A 
/ 
d 
G 
l 
j 
(2) 
a 
/ 
D 
b 
c 
K 
9 
l 
J 
h i E 
(3) 
A 
/ 
B 
G 
k 
G 
l 
H 
I 
e 
(4) 
A 
F 
D 
b 
C 
k 
G 
L 
J 
h i E 
(5) 
A 
F 
D 
B 
K 
G 
L 
J 
H 
E 
§ 6. Skew Amphicheirals of the Second Order. 
In a note added to his last paper on knots, Tait gives a special knot* of order 
8 which he classes as an amphicheiral of the second order ; although, strictly 
speaking, it does not belong to the second order, since corresponding compartments 
are not opposite when fitted on a sphere. 
In the investigation of amphicheirals with twelve crossings this type assumes 
* Trans. Roy. Soc. Edin., xxxii, p. 500 ; Scientific Papers, i, p. 342. 
