
160 PROFESSOR TAIT ON KNOTS. 
which itself presents very grave difficulties, at least where n is a composite 
number. In fact it is probable that the solution of these and similar pro- 
-blems would be much easier to effect by means of special (not very complex) 
machinery than by direct analysis. This view of the case deserves careful 
attention. : 
In a later section it will be shown how, by a species of partition, the various 
forms of any order of knottiness may be investigated. But we can never 
be quite sure that we get all possible results by a semi-tentative process 
of this kind. And we have to try an immensely greater number of par- 
titions than there are knots, as the great majority give links of greater or less 
complexity. 
§ 13. But even supposing the processes indicated to have been fully carried 
out for 8, 9, and 10-fold knottiness, a new difficulty comes in which is not met 
with, except in a very mild form, in the lower orders. For when a knot is 
single, 7.¢., not composite or made up of knots (whether interlinked or not) of 
lower orders, any deviation from the rule of alternate + and — at the crossings 
gives it, in general, nugatory crossings, in virtue of which it sinks to a lower 
order. But when it is composite, and the component knots are separately 
irreducible, the whole is so. Thus there are more distinct forms of knots than 
there are of their plane projections. For instance, the first species (a) of the 6- 
fold knots (§ 8) may be made of three essentially different forms, for the 
separate “trefoil” knots of which it is made may (when neither is nugatory) be 
both right-handed, both left-handed, or one right and the other left-handed. 
This species is thus, from the physical point of view, capable of furnishing 
three quite distinct forms of vortex-atom. And it will presently be shown 
that in each of these forms it is capable of having regular alternations of + and 
—, or a set of sequences at pleasure. 
At least one knot of every even order is amphicheiral, 1.e., right or left-handed 
indifferently, but no knot of an odd order can be so. Hence, as there is but one 
3-fold knot form, and one 4-fold, there are two possible 3-fold vortices, right 
and left-handed, but only one 4-fold. A combination of two trefoil knots gives, 
as we have seen, three distinct knots ; that of two 4-fold knots would give an 
8-fold, with only one form. When a 3-fold and a 4-fold are combined, as in 
Class I. of § 10, there are two distinct vortices, for the trefoil part may be right 
or left-handed. Thus it appears that though we have shown that there are 
very few distinct outlines of knots, at least up to the 7-fold order, and though 
probably only a very small percentage of these would be stable as vortices, 
yet the double forms of non-amphicheiral knots give more than one distinct 
knot for each projected form into which they enter as components. 
