598 
MARY GERTRUDE HASEMAN ON AMPHICHEIRAL KNOTS. 
defined as a mid-point of a lap of the thread so located that corresponding crossings 
occur at equal arcual distances when the knot is transversed along this thread in 
opposite directions from the point. Let the amphicheiral centre (p 1 be the mid-point 
of the lap of thread between the two corresponding crossings y and q of a knot 
with n crossings, and denote by a 1; f3 1 the number of pairs of crossings which elapse 
before the next occurrence of p, q respectively as the knot is traversed from p to (pi 
through q. Hence n— 1 — cq, n — 1 — Pi will be the number of pairs of crossings 
which elapse before the next occurrence of the crossings p, q respectively as the 
knot is traversed from q to (pi through p. By the definition of an amphicheiral 
centre, oq = n— 1 — fii or a 1 -\-fi 1 = n — 1. Similarly + $ = n — 1 , where and ft, are 
two crossings of the intrinsic symbol at equal arcual distances from ^i- Thus an 
amphicheiral knot of the first order, as well as its pairs of amphicheiral centres, may 
be detected very easily from its intrinsic symbol. For example, the intrinsic symbol 
444848 - 5 9 5 9 9 9 4 • 9 4 4 4 8 4 8 ■ 5 9 5 9 9 9 4 • 9 
a f b k c l d n e a f b g c h m i d j e k g l h m i n j 
exhibits one pair of amphicheiral centres between the crossings l, d and e,k, as well 
as a second pair between g, c and n, j. 
On the other hand, from Tait’s # definition of an amphicheiral of the second 
order, it is seen that every crossing is separated from its correspondent by n — 1 pairs 
of crossings, and hence the first n numbers of the intrinsic symbol are identically 
equal to the remaining numbers, the sequence of numbers being the same. Accord- 
ingly the knot whose intrinsic symbol is shown in (1) on page 1 is an amphicheiral 
of the second order. 
§ 2. A New Construction for the Amphicheirals of Order 1. 
The intrinsic symbol for all of the amphicheiral knots of the first class of 
orders 1 and 2, as constructed by Tait f is arranged in one of the two sequences 
stated above, whereas this is not the case for all of the amphicheirals of the second 
class of orders 1 and 2. However, in a census (M. G. Haseman, Trans. Roy. Soc. 
Edin., vol. lii, p. 253) of the amphicheiral knots with twelve crossings, it is found 
that the form shown in fig. 3, which is obtained by the unsymmetrical distortion 
D]" of the form shown in fig. 2, and belongs therefore to the second class, 
possesses the intrinsic symbol, 
177 10 44177 10 44177 10 44177 10 44; 
this classes it among the amphicheirals of the second order. But it is impossible to 
put this knot on the sphere so that corresponding compartments are opposite ; that 
is to say, it cannot be constructed in the plane by means of a great circle and a pair 
of twin circuits. Hence it cannot belong to order 2 as defined by Tait.J It is 
* Tait, Trans. Boy. Soc. Edin., vol. xxxii, p. 497. f Ibid., Plate LXXIX. J Ibid., p. 497. 
