PROFESSOR TAIT ON KNOTS. 189 
And a little consideration shows that on the following pattern may be 
formed amphicheiral knots of all the orders included in 6” and 4+ 6n :— 
CQ 
Among them these forms contain all the even numbers, so that there is 
least one amphicheiral form of every even order. 
Many more complex forms may easily be given. See, for instance, Plate 
XVI. figs. 18, 19, 20. Some are closely connected with knitting, &. 
An excessively simple mode of obtaining such to any desired extent is to 
start with an amphicheiral, whether knot or link, and insert additional crossings. 
These must, of course, be inserted symmetrically in pairs, each in the original 
figure being accompanied by another which will take its place in the perversion 
or image. 
_ Thus, taking the simplest of all amphicheirals, the single link (Plate XVI. 
first of figures 27), we may operate on it by successive steps as in the succeeding 
figures. 
The second, third, and fourth are formed from the first by adding, the fifth 
and sixth from the fourth by removing, pairs of crossings. The third, like the 
first, is a link ; the others are knots. : 
Figures 28, Plate X VI., give another series, of which the genesis is obvious. 
The protuberances put in the first figure, for instance, show how it becomes 
the second. The fifth of fig. 27, and the second and fourth of fig. 28, all alike 
represent the amphicheiral form (8) of § 8. But we need not pursue this 
subject. . 
§ 48. It will be seen at a glance that the first pattern in last section gives 
for two loops (7.¢., four crossings) the knot of § 6; while the third pattern as 
drawn is simply 6 of § 8. Inthis form of the knot, the two dominant crossings 
(§ 46) are those in the middle, and mere inspection of the figures shows that 
the whole knotting becomes nugatory if the sign of either of these be changed. 
It might appear at first sight that amphicheirals of the same knottiness, 
formed on such apparently different patterns as the two first of last section, 
would be necessarily different. But the very simplest case serves to refute this 
notion. For the lowest integers which make 
4n=24+6n’ 
give 8 as the value of either side. Figs. 22, 23, Plate X VI., represent the corre- 
sponding amphicheirals, apparently very different, but really transformable into 
one another by the processes of § 11. Fig. 21, Plate X VI., represents another 
8-fold amphicheiral form, suggested by a somewhat similar pattern. I hope to 
VOL. XXVIII. PART I. 3 C 
