MARY GERTRUDE HASEMAN ON AMPHICHEIRAL KNOTS. 
601 
Another very good example (see fig. 6) of a skew amphicheiral is given by 
the symbol 
1 3 3 22 1 8 8 22 20 20 15 15 1 3 3 22 1 8 8 22 20 20 15 15 
a e b a c l d c e b fvgkligirj i k h Id 
1 3 3 22 1 8 8 22 '20 20 15 15 1 3 3 22 1 8 8 22 20 20 15 15 
m q n m ox p o q n r j s w t s u f v u w t x p. 
Because of the relation of the curve c 2 r to the curve c x the crossing a may 
correspond to either g or s ; likewise the crossing m may correspond to either g 
or s. Therefore we may expect not only the numbers in the last half of the 
sequence to be a repetition of those in the first half, but also the first half to 
consist of a repetition of a certain group of numbers ; the number of repetitions 
will probably depend on the number of contacts. 
The only skew amphicheirals, which I have found, may be obtained as the 
unsymmetrical distortions of an amphicheiral of the first class of order 1, and 
in view of the fact that the curve for knots with 4, 6, 8, 10, 12, 14 crossings 
seems to lead always to a knot which can be distorted into an amphicheiral of 
the first class of order 1, I am of the opinion that they do not constitute a 
■distinct class, although it may be that they will form another class in the case 
of the knots with a greater number of crossings. 
If, therefore, an amphicheiral knot is defined as one whose primary and secondary 
symbols are identical — that is to say, one whose intrinsic symbol belongs to one of 
the two arrangements mentioned on p. 598 — it is seen that Tait’s classification given 
in Trans. Roy. Soc. Edin., vol. xxxii, p. 499, is sufficient provided that it be 
admitted that an amphicheiral knot can belong to the first class of one order and 
to the second class of the other order. 
§ 4. Amphicheiral Knots of Order 2 with Fourteen Crossings. 
As has been shown by Tait, Trans. Roy. Soc. Edin., vol. xxxii, there are no 
amphicheiral knots of order 2 with 4, 6, 8, or 10 crossings. There are two knots * 
of the second order with twelve crossings, both of which may be constructed on 
models involving one pair of contacts, although they appear among the knots 
which required a greater number of contacts. In a consideration of the maximum 
number of contacts necessary to construct the amphicheirals with n crossings I 
was led to construct the amphicheirals of the second order with fourteen crossings, 
of which there are ten in number, as shown in Nos. 23-32 in the Plate. Nine of 
these were constructed on models with one pair of contacts, whereas the tenth 
one, No. 32, required two or more pairs of contacts. Hence it will be necessary 
to pass to the amphicheirals with a greater number of crossings in order to 
determine the maximum number of contacts required. 
An interesting amphicheiral is the form which is obtained by the single distor- 
* M. G. Haseman, Trans. Boy. Soc. Edin., vol. lii, p. 254. 
TRANS. ROY. SOC. EDIN., VOL. LII, PART III (NO. 23). 
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