774 



PROFESSOR C. N. LITTLE ON 



Lastly, suppose a class VI. form to have the sequence of three of fig. 1. The 

 parts A, B, C, D, a, b, c, d must all be distinct, and A, B, a, b can not 

 C 1 a | B | d be 2-gons, or the form would be reducible. Nor can the latter be 



c | A | b 



Fig. 1. 



D 



3-gons. 



cU/b 



X b :-A 



Fig. 2. 



Fig. 3. 





F 



f 





c 



a 



b 



d 



c 



A 



b 



D 











Fig 4. 



For, suppose any one, as A, to be a 3-gon. If, now, the 

 thread be shifted from the position fig. 2 to that of fig. 3, the 

 part A is lost to the leading partition and the form comes 

 under class V., which, as has just been shown, can have 

 no form with three consecutive overs. Hence we have the 

 portion of the class VI. tenfold shown in fig. 4. A moment's 

 consideration shows that the parts marked are all distinct. 

 Of the three pairs of adjacent parts C, c ; D, d ; F, f, only one in 

 each can be a 2-gon. Hence to complete the form at least two 

 additional crossings must be introduced, so that the form becomes 

 at least an elevenfold. Still less is it possible for A, B, a, or b 

 to have more sides than four. Hence no class VI. tenfold can 

 have a three sequence. 

 This completes the proof of the theorem that no reduced non-alternate knot with 

 three consecutive overs has fewer than eleven crossings. 



It is easy to verify these theorems by inspection of the plates of alternate forms. 

 An irreducible elevenfold knot form with three consecutive overs is shown at B, the 

 last in PI. III. As alternate it is No. 39 of my census.* 



8. Twist. — Let the direction of the moving point, continuously tracing the 

 projection of a knot be recorded at each crossing by marking the thread with 



arrows. Crossings are of two kinds, as shown in the 

 figs. 5 and 6. I call the first a twist of 4- x, the 

 second a twist of — -k. 



Theorem. — The total twist of a reduced knot is 

 constant for all forms in which the knot can be pro- 

 jected. The proof is very simple. The twist of the 

 crossings is not altered by any of the transformations permissible to alternate forms, 

 since these consist of rotations of a portion of the knot through an angle of it 

 about an axis in the plane of the knot projection. This does not change the relation 

 of the arrows to a crossing nor a A crossing to a y, or vice versa. In the changes 

 peculiar to non-alternate forms the thread is shifted from one portion of the knot 

 to another, so as to alter the position of two consecutive overs (or unders). Either 

 the twist of the two crossings is unchanged, or else the twist of the two crossings 

 was originally unlike, and in both crossings it is reversed. 



9. Knots may be classified according to twist. The non-alternate tenfolds will be 

 so classified in the following census. 



10. An amphicheiral knot is one that can be distorted into its own perversion; and 



* Trans. R. S. E., vol. xxxvi. pi. I. (39. D x . /*.) 



Fig. 5. 



Fig. 6. 



