



188 PROFESSOR TAIT ON KNOTS. 
one by one, and since (§ 20) the whole number of compartments is greater by 2 
than the number of crossings, the number of crossings must be even. Let it 
be 2n, and let p,, Po, .--.- Pn4i be the partitions. Then our selection must be 
made from the numbers which satisfy 
D4 Py see ee O45 = 40, 
no one being greater than the sum of the others. If a set of such can be 
grouped as in § 20 so that the other set for the complete scheme: shall be the 
same numbers with the same grouping, we have an amphicheiral form. The 
words in italics are necessary, as the following example shows; for here the 
black and white compartments have the same set of partitions but not the 
same grouping, and the knot is not amphicheiral:— 
G3 
But a different grouping of the same set of partitions gives the amphicheiral 
form below 
(eo 
But an easier mode of procedure, though even more purely tentative, is the 
following :—If a cord be knotted, any number of times, according to the 
pattern below, 
= 
it is obviously perverted by simple inversion. Hence, when the free ends are 
joined it is an amphicheiral knot. Its simplest form is that of 4-fold knottiness. 
All its forms have knottiness expressible as 42. 
The following pattern gives amphicheiral knottiness 2+ 6 :-— 
ge 
