PROFESSOR TAIT ON KNOTS. 183 
Plate XV. figs. 16 and 17, which have been already described (§ 27). <A full 
investigation of the higher knottinesses of this class (especially when fully 
beknotted) would well repay the trouble it would involve. 
As they are all amphicheiral, and in each case the crossings are divisible 
into two sets, those of each set being in all respects alike, while those of different 
sets differ only as to silver or copper, it is no matter (so far as testing be- 
knottedness is concerned) which crossing we suppose to have its sign changed. 
In the 8-fold amphicheiral of fig. 16 the change of any one sign reduces the 
whole to the irreducible trefoil knot (§ 16), right or left-handed according as we 
have changed one of the four outer, or of the four inner, crossings in the figure. 
Hence it has two degrees of beknottedness. But if we change the signs of one 
set of crossings (Plate X VI. fig. 24) so as to make all the crossings alike silver 
(or copper), we find the knot irreducible, though with continuations of sign ; 
but with three degrees of beknottedness. And it is easy to see that it can now 
be analysed into two right-handed trefoil knots linked together as shown in the 
other part of the figure. But the linking is /e/t-handed. Wad it been right- 
handed we should have had + and — alternately, and thus we could not have 
transformed back to the form with continuations of sign (§ 4). 
Similar remarks apply to the 10-fold amphicheiral plait (Plate XV. fig. 17). 
_ Change of any one sign reduces it to the third form of 6-fold knottiness (y, § 8), 
which has only one degree of beknottedness. Hence the 10-fold plait has but 
two degrees of beknottedness when its signs are alternate. If we make all its 
crossings silver (or copper), as in Plate XVI. fig. 25, it has jour degrees of be- 
knottedness ; and the reason is obvious from the other half of the figure, where 
it is seen to be made up of a pair of irreducibles—a pentacle and a trefoil, once 
linked together. Thereis one degree of beknottedness for the trefoil, one for the 
link, and two for the pentacle. The trefoil and pentacle are right-handed, the 
link left-handed, else we should not have had the continuations of sign which 
the figure must show. 
A very curious illustration of this is to be found in the excepted cases, where 
the number of crossings is a multiple of six. From the two figured (Plate XV. 
figs. 15, 18) it is obvious that all of these are formed by three unknotted closed 
curves, no two of which are linked together, yet the whole is irreducible, having 
alternate signs. Hence we require a third term to complete our descriptions— 
knotting, linking, locking (¢). 
To give the greatest amount of belinkedness to these figures, let us suppose 
the ovals taken all the same way round, and arrange so that all the crossings 
shall be silver. Then we have continuations of sign (Plate XVI. fig. 26) as in 
the knots of the same series. But whereas Plate XV. fig. 15, if made of wire, 
is particularly stiff, the new figure is eminently flexible. ‘This seers to have 
been practically known to the makers of chain armour. 
