252 
MARY GERTRUDE HASEMAN : ON KNOTS, 
fact that it is not obtained by means of a twin circuit in contact with a circle ; 
that is to say, corresponding compartments are not opposite. 
Thus it is seen that Tait* is not justified in stating that there are but two 
possible ways in which corresponding spherical compartments of a knot may be 
equal. Further, contrary to Tait,! it is possible for an amphicheiral knot to 
belong to the first class of one order and to the second class of another ; as, for 
example, the skew amphicheiral (No. 61), which belongs to the second class of 
the amphicheirals of the first order, and first class of the skew amphicheirals of 
the second order. 
The explanation of the difficulty is that the set of X compartments cannot be 
moved on the sphere without affecting the set of <5 compartments ; hence it is not 
always possible to place the whole knot in one of the two ways considered by Tait. 
Inasmuch as all possible cases of amphicheiralism do exhibit themselves in the 
compartment symbol, it seems highly advisable to use this in the definition, which 
may then be formulated as follows 
Definition . — An amphicheiral knot of the first class is one whose primary and 
secondary symbols are identical as to numbers and arrangement, but with rotation 
in the same sense for those of the first order, in the opposite sense for those of the 
second order. Any form obtained from an amphicheiral of the first class by non- 
conjugate distortions \ is an anfphicheiral of that same order, but of the second class. 
It is, however, possible for a knot to belong to the first class of one order and to 
the second class of the other. Since the two symbols are alike, it is possible by 
a deformation to replace the amplexus with the corresponding compartment 
and the perversion is obtained. 
With one exception (No. 61) the skew amphicheiral with twelve crossings turn 
■out to be amphicheirals of the first order. This overlapping of the different 
divisions has been detected by Tait for knots with ten crossings, where there are 
no amphicheirals of the second order that are not also of the first order. As shown 
here for twelve crossings, there are some of the second order not included under 
those of the first order ; presumably with a greater number of crossings there may 
exist amphicheirals of the second order that escape any of the other divisions. 
This completes the census of the twelvefold amphicheirals, of which there are 
sixty-one, as compared with one fourfold, one sixfold, five eightfold, and thirteen 
tenfold amphicheirals. 
* Tait, Trans. Roy. Soc. Edin., xxxii, p. 498 j or Scientific Papers, i, p. 340. 
■j- Tait, Trans. Roy. Soc. Edin., xxxii, p. 499 ; or Scientific Papers, i, p. 341. 
| It must be remembered that if a knot is amphicheiral of the first order, with more than one pair of centres, 
distortions that are non-conjugate for one pair may be conjugate for another pair. 
