CONSTRUCTION OF KNOTS OF FEWER THAN TEN CROSSINGS. 



297 



For 3 upon 5, 



3 A/on 5 A gives 8 Ad; 



3 Afe on 5 A gives 8 A/ and 8 Ae ; 



3 Afe on 5 B attempted gives 8 Z 



erroneously, or leaves a concurrence. h Affc on 5 A gives 8 Ag, by arts. 16, 17, 

 and 8 A^, art. 27. 



For 4 upon 4, 



Jiff on 4 A gives 8 Ai ; 



4 A/eon 4 A „ 8 A/; 



4 Aee on 4 A 



» 8 



AJc and a AL 



The janal zoned poles on 8 Ai are two flaps, two edges of the 4-gon, and two 



opposite not plane 6-gons. The two poles of 8 A/ are a flap and an edge 33 : 



on 8 Ak the janal zoneless poles are edges 33 : on s Al are two pairs (66) 



and (33) of janal zoneless 2-ple poles ; and a third pair are two 6-gons not 



plane. 



There are two constructions by the charge ^Aee, because neither e nor e 



(art. 27) is zoned polar. 



For 2-2 on 4 (art. 38), 



4 A 2 /c on 4 A gives 8 Am, 



which has all the symmetry of 4 A ; the four 2-ple zoneless janal poles are where 

 they were, and the zoned janal poles are the two flaps of the imposed charges. 

 Finally, 



J?ffc on 4 B gives 8 An, 



having a janal zoned pole in each flap, and another pair in opposite non-plane 

 6-gons. 



8 Ap is the only solid knot of eight crossings, a 4-zoned monarchaxine 

 homozone, whose eight identical janal 2-ple zoneless poles bisect eight polar 

 edges 33. Thus we have constructed seventeen unsolids without concurrence, 

 . 8 X 8 Y . . . 8 An, of which nine are unifilar. 



46. We complete our list of unsolids by art. 29 — 



7 A gives 



! 8 A ?; 



7 B » 



8 Ar, 8 As ; 



7 C , 



s At ; 



7D „ 



8 Au ; 



7 E „ 



8 Av, 8 Aw, 8 Ax 



7E „ 



& Ay, 8 Az ; 



7 (j „ 



8 Ba, 8 B6 ; 



7H „ 



8 Bc, 8 Bd ; 



J 



3 Be, 8 B/ 



7 J gives 



8%. 8 B/i J 



6 A „ 



8 B *> 8 B / ; 



6 B „ 



8 BJc, 8 Bl, g Bm, 8 Bn ; 



6^ » 



8 Bp, gBj ; 



6^ „ 



8 Br, 8 Bs ; 



6 E „ 



8 Bt, 8 Bu ; 



5 A „ 



8 Bv, 8 Bw ; 



4A „ 



8 Bx, 8 By, 8 Bz ; 



5A ,, 



S C(X. 



Of these 36 we have figured only half, the 18 of them which are unifilar ; 



VOL. XXXII. PART II. 3 C 



