CONSTRUCTION OF KNOTS OF FEWER THAN TEN CROSSINGS. 



305 



60. We have next to lay 4 and 2 -2 upon 5, 



4 A/on 5 A gives 9 D/; 



A e f 0n 5 A . 



. £>9\ 



4 A/e on 5 A , 



, Q Dh; 



4 Aee on 5 A „ 9 Di . 



4 A 2 j/e on 5 A gives 9 D/ (art. 38). 

 4 A 2 / C on 5 B „ ,m. 



5 B was made (art. 31) by adding a concurrence to 4 A ; but it is also 4 A/fc 

 on 3 A, though improperly made (art. 38) ; and as we have two charges to im- 

 pose, we can both spoil the concurrence and cover the least marginal of 5 B, 

 thus making 9 D&. 



61. Finally, we have to lay 2'3 and 3*2 upon 3, 



5 A?ffc on 3 A gives 9 DZ and 9 Dm. 



z A?ff on 3 A gives 9 Dw. These lay 2 - 3 on 3. 



4 A 3 /c on 3 A gives Dp. This lays 3-2 on 3. 



In 9 D/, 9 Dm, and 9 T>n the symmetry is maintained about one of the 2-zoned 

 axes of 3 A, though not a 2-zoned symmetry. 



We have constructed 115 9-fold knots without concurrences, of which 63 

 are subsolids, and 52 are unsolids without concurrences. Among the 63 are 

 30 unifilars, and among the 52 are 25, making 55 unifilars without concurrences. 



62. There remain only the 9-fold unsolids which have concurrences. 

 The number of ways in which a knot K' of n-c crossings can be made by add- 

 ing c concurrences of flaps into K of n crossings is easily seen when the 

 symmetry of K' is given, K' having no concurrence — 



8 V gives 9 Fc , 9 Fd , 9 Fe ; 



8 A gives 9 D<7 







8 B „ 



9 Dr; 







8 c „ 



9 Ds , 



9 T>t; 





8 D » 



9 ~Du; 







8 E „ 



9 T)v , 



9 Dw 



j 9-Lte j 



8 F „ 



9%- 



9 T>z; 





8^" » 



9 Ea , 



9 EZ>, 



9 Ec; 



8 H „ 



gEd; 







8 I - 



9 Ee, 



*W, 



9%; 



8 J „ 



9 E7i; 







8 K „ 



9 E* , 



9 Ey, 



■Bfc; 



8-L » 



9 E£ , 



9 Era 



> 



8 M „ 



9 E% 







8 N „ 



9 E i> 







8 P ,, 



9 E ? 



» 9 Er ; 





8 Q >, 



9 Es, 



9 E£; 





S R 



9 E« 



, 9 E« 





8 s „ 



9 Eiw 



> Q^ X 



j 



8 T „ 



9%. 







8 U „ 



9 Es , 



9 Fa, 



9 F&; 



8 w 

 8 x 



8 Y 



8 z 



8 Aa 



8 A& 



8 Ac 



8 Ad 



8 Ae 



8 A/ 



s A # 

 s Ah 



8 Ai 



8 Ay 



6 Al 



8 A??i 

 Q An 



•f/; 



9 % , fl Ffc ; 



9 Fi, 9 F/, 9 Fft; 

 9 FZ , 9 Fm ; 

 9 ¥n , 9 Fp ; 

 9 F ? , 9 Fr , 9 Fs ; 

 9 Yt , 9 F% , 9 Fv ; 

 9 Fw , 9 F« ; 

 9*V , 9F2 ; 

 9 Ga , 9 G& , 9 Gc ; 

 qGcI ', 

 9 Ge; 



9 G /5 



fig , 9 G/t ; 

 9 Gi; 



9G/; 

 9 G# ; 

 9 G£ , 9 Gm . 



VOL. XXXII. PART II. 



3D 



