CONSTRUCTION OF KNOTS OF FEWER THAN TEN CROSSINGS. 291 



This is read thus — the flap on 5a has for collaterals 3 and 4, and for coverticals 

 3 and 4 ; so has the flap on 12 : which leads ? 

 Next, conceiving 56 drawn, we write, 



(56)=43,44; (12) =44,43; (43)=43,44. 



Here by art. 19 (12) appears to be leader, until we observe that it is fixed by- 

 art. 22, and cannot be a competitor. 

 We therefore write more correctly, 



(56) = 43,44 > (43) =43,44? (12) is fixed; 



which inquires, Does (56), which is 43,44, lead (43), which is also 43,44? 

 We consider this second as well as the preceding note of interrogation a pre- 

 sumption of symmetry (art. 21). 



Drawing the flap (5a) we obtain 6 A, and the flap (56) gives us 6 B, on both 

 of which the leading flap so drawn is marked 56. Observe that in our figured 

 subsolids of n crossings, the leading flap is always marked n(n - 1). Our pre- 

 sumptions of symmetry are verified in the two results 6 A and 6 B. 



The polar edges of the heteroid zoneless axis of 6 A are evident in the figure. 

 The two-zoned axis of 6 B has for faces a flap and an opposite 4-gon. The 

 other two flaps of 6 B are like epizonals. 



It was possible to connect by a flap the two edges of the flap 43 in 4 A. But 

 this could have completed an unsolid having a linear section through 3 and 4 ; 

 and completed it wrong, because no unsolid is ever made by so adding a flap. 



33. We next take the unsolid 4 B, considering whether or no a flap can be 

 drawn on it to make it a subsolid of six crossings. Eeadily we perceive that 

 by joining two opposite flaps we can both spoil the concurrence and block the 

 linear section. This gives us 6 C, which has all the symmetry of the wedge which 

 it becomes when an edge is removed from every flap. The three, 6 A C B and 6 C, 

 are all the subsolids of six crossings. 



34. We seek now the unsolids of six crossings. To obtain them by least 

 marginal charge or charges (art. 25), we have to lay 2 upon 4 and 3 upon 3. 



There is but one charge that can add two crossings only, ^Affc, which means 

 4 A imposed (art. 15), by the section Jfc. Imposing this on 4 A we get 6 D, a zoned 

 triaxine, whose three janal 2-zoned axes have for poles, one two tessaraces, 

 a second two flaps, and the third two 4-gons symmetrical but not plane, which 

 have two common summits and no common edge. 



Laying next on 3 A the charge z Aff (art. 15), we obtain 6 E, another zoned 

 triaxine, whose janal poles are two edges, two tessaraces, and two hexagons 

 alike and non-planar. There is in truth no least marginal subsolid in either 

 6 D or 6 E, the two halves of the knot being identical in each. But it is instruc- 



