772 PROFESSOR C. N. LITTLE ON 



eminently of Professor Tait, the theory of the alternate knot is well understood. It 

 may be useful to recapitulate the main points of this theory : 



(a) The knot is reduced (projected with fewest crossings) if no one of the compart- 



ments into which its projection divides the plane is opposite itself. 



(b) Reduced forms of the same knot divide the plane into two sets of vertically 



opposite parts, with a constant number of parts in each set. This gives a 

 convenient basis for classification. 



(c) It is a simple problem in the circular arrangement of letters to determine from a 



given reduced form all reduced forms of the knot. Hence it is easy to say as 

 to two given forms whether or not they are projections of the same knot. 



(d) Simple methods are known by which all the knots of a given order (minimum 



number of crossings) can be found. 



(e) The theory of amphicheiral knots. 



5. It is quite the contrary with the non-alternates. They constitute an almost 

 untouched field, bristling with difficulties. In these Transactions, vol. xxxv. part ii. 

 p. 664, I published a census of these knots for Orders Eight and Nine ; for brevity I 

 shall refer to this paper under the letter A in brackets. 



6. It is there stated, [A] § 8, There is no reduced non-alternate ± knot of fewer 

 crossings than eight. I proceed to give formal proof*: — 



In class II., order n, all knots are obviously alternate. 



In class III., order n, the leading partition (set of compartments with smaller number 

 of parts) has three parts, say A, B, C. If any one of the connections (AB), (BC), (CA) 

 is null, the form is not a knot. If any one is a single crossing the knot is alternate, 

 [A] § 3. If two of them have each two crossings there must be a link, that is, at least 

 two threads. Hence there must be for these connections at least 3, 3, 2 crossings 

 respectively, and a non-alternate knot of class III. must be at least an eightfold. 



Consider class IV., order n. Here there are in the leading partition four parts, 

 A, B, C, D, and these have six connections. 



First, let any one, say (AB), be null. Then in order that the form may be the pro- 

 jection of a reduced knot, (AC), (AD), (BC), (BD) must exist. There are two cases : — 



(a) Let (CD) also be null. If now any one of the existing connections be a siugle 

 crossing, say X, then all of the crossings of the other connections will also be X, by [A] 

 § 3, and the form alternate. But if not, the form is at least elevenfold, since only one 

 connection can consist of an even number of crossings, else the form would be a link. 



(b) Let (CD) exist. We now have D joined to C by B, by A, and by the one or more 

 crossings of (CD). If either (DB) or (CB), and at the same time either (DA) or (CA), 

 are single crossings, the form is alternate. If (DB) and (CB), or (DA) and (CA), arc 



* The classification of alternate knots according to the number of parts in the projection, in that set of vertic'llj 

 opposite compartments which has the smaller number, does not answer for non-alternates, since the same knot can be 

 projected in forms belonging to different classes. Later, in § 9, a new basis of classification will be proposed. In this 

 and the following sections, however, the term class has the old signification. 



