CONSTRUCTION OF KNOTS OF FEWER THAN TEN CROSSINGS. 299 



For the rest, 



(54) leads (fb), (aa)=U and (a5)=44, 

 (67) leads (ab) = 35, (ea) and (cb). 



We have to draw (bb) = oS, (act), (fc) and (£5) = 44, expecting symmetry 

 with the last, 



(56) =53 gives 9 E ; 

 C/ c ) » 9 F ; 



(66) =44 gives 9 G ; 

 (aa) , 9 H ; 



whose symmetry and circles are on their figures. 



49. 7 C annexed has monozone faces 1357, 12467, 354, and asymmetric 

 faces 456, 567, and the flap. 



The different lines on it are, 



db, ac, cc, each 36 ; 

 aa, ac, be, each 45 ; 

 bd, ee, each 44 ; 

 ed, eb, each 35 ; 



besides lines 34 that can be drawn in triangles. 



(aa) has no rival : 

 in la5, (a5)=63,53>(67) = 63, 53 ? 



in 2ac, (ac) = 63, 43>(67) = 63, 43 ? 



in24ca (ac)= 54>(67) = 53; 



inl2c6 (6c) = 54>(67) = 53 or (12)=43 ; 



(cc) = 63, 44 > (12) or (67) each = 63, 44 ? 



For the rest, (12) or (67) leads them all, as well as all flaps on lines 34. 

 We have to draw six flaps, looking for symmetry with three, 



(ab) gives us 9 J ; 



(ac) „ 9 L in 24ca. 

 (cc) „ 9 N. 



The symmetry and circles of these are on the figures. 



50. Next comes 7 D here drawn. The polar 4-gon is 5217, and the asym- 

 metric faces are 5234, 123, 143. The only different 



lines to be drawn are, 



fd,fh, ch, cd, dg x in ldg, and dg 7 in 7dg, all 53 ; 



fc, dd, dh, gg, all 44 ; 



eg, ei, gi, he, hi, ci, all 43 ; 16 lines. 



(fd) = 53 > (67) =43; 



(fh) =53>(67)=43; 



(cd) = 53, 54 > (32) = 53, 53; (67) = 43; 



(d&) = 53, 54 > (67) = 53, 53, or (32) = 44 ; 



(fa) = 53 > (32) = 43 ; and (67) is axed (art. 23) ; 



(aa) gives us 9 T ; 

 In 2ac {ac) „ 9 K ; 

 in 216c (6c) „ 9 M ; 



