CONSTRUCTION OF KNOTS OF FEWER THAN TEN CROSSINGS. 



301 



We have eleven flaps to draw, with five queries about symmetry, which 

 speedily reveals itself, 



(¥) gives 9 AZ> ; 

 (ed) „ 9 Ac ; 

 (ce) „ Q Ad ; 

 (be) „ 9 Ae ; 



(&e) » 9 4/; 



( c /) » 9%; 



(/rf) gives A7i 



(Im) 

 (dk) 

 (if) 



» 9 



9 4/ 



,A/fc 



A/. 



The symmetry and circles of all are written on their figures. The poles of 

 9 A/ are the tessarace common to the two 5-gons, and the flap which connects 

 them : those of 9 A/c are a 6-gon and a 4-ace. 



52. The sixth and last subsolid is 7 F, which has only one asymmetric face 

 (1567), and one asymmetric flap (67). The flap (25) is 



epizonal. Eighteen different flap-lines can be drawn : , e_ 



ae, b'd, b'e, of, be, ef, ec, fd, all 53 (bj and be clotted below) ; 



ac, bb, b'f, ee, ed, cj, all 44 ; n 



f, a'/, dd, dj, all 43. 



(6&) = 44>(25) = 43; 



(b'd) in Vd7=53>(2o)=43 and >(34)=44. 



For the rest, 



(25) leads b'e, b'f, cj, ed ; 



(34) leads ae, ee, ec, ac, a'f, dd, and dj in *ldj ; 



(67) leads fd, be in 2>bc,ff,fe, and bj in 4bj. 



We have two flaps to draw, 



(bb) giving 9 Am and (b'd), giving g An, 



whose circles and symmetry are read on the figures. 



53. We betake ourselves next to the unsolids of seven crossings, which, by 

 a leading flap, can become subsolids. These are 



r G, 7 H, 7 I, 7 J, 7 K, 7 L, 7 M, 7 N, 7 Q, 7 R, 7 S 



On 7 G annexed can be drawn to block the linear section only three lines, 

 which are 



(a&) = 54>(12) = 43; 

 in cVl, (cb) = 63, 44 > (54) = 63,43 and > (12) = 43 ; 

 in cbll, (c&)=54>(54) = 53 and >(12) = 43. 



We draw three flaps, 



(ab) giving 9 Ap, (cb) giving 9 A.g, and (cb) giving 9 Ar ; 



whose description is seen on their figures. 



