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MARY GERTRUDE HASEMAN ON AMPHICHEIRAL KNOTS. 
It is. possible to have the two curves and C 2 in their various positions intersect 
in or' pairs of points, although it is not always possible to make corresponding laps of 
the thread intersect without introducing extra crossings. By this process I have 
succeeded in constructing all of the amphicheirals of order 1 with four, six, eight, 
and ten crossings (Nos. 1-21 in the Plate*), and find the tenfold knot No. 21 in 
the Plate, omitted by Tait in his census (Trans. Roy. Soc. Edin ., vol. xxxii, 
Plate LXXIX). 
Likewise the knot No. 22 in the Plate has been omitted from my census (see 
Trans. Roy. Soc. Edin., vol. lii, pp. 253-4) of the amphicheirals with twelve crossings. 
It is possible that this method will reveal other omissions. It is to be noted that the 
maximum number of contacts were used in the constructions of these amphicheirals, 
Fig. 4. 
Fig. 5. 
Fig. 6. 
but 1 cannot say whether this is necessary in the construction of the knots with 
a greater number of crossings. 
§ 3. Skew Amphicheirals. 
If, however, the curve c 2 / , obtained by a single reflection in the plane is used 
instead of the curve C 2 , there results upon rotation a knot which can be distorted 
into an amphicheiral of the first order— that is to say, it belongs to Tait’s second 
class. When the curve Cj is symmetrical about the line pip%, the resulting knot 
is an amphicheiral of the first class of order 1, since then the curve is identical 
with the curve C 2 . 
In this construction there arise certain knots, called by me skew amphicheirals 
of the second order, which exhibit the amphicheiral symmetry in spite of the fact 
that they belong, by the above statement, to the second class of order 1. The 
intrinsic symbol of all such knots, as I have found, classes them with the amphi- 
cheirals of the first class of order 2, although corresponding regions are not opposite 
on the sphere. An example of a skew amphicheiral is shown in fig. 5 ; it is found 
to be identical with the knot in fig. 3. It is to be noted that in the knot shown in 
fig. 5 the curve possesses symmetry of such a nature that its relation to curve c x 
is the same whether c 2 ' be rotated to the right or left. 
■* The numbers at the lower right-hand corners are Tait’s numbers. 
