CONSTRUCTION OF KNOTS OF FEWER THAN TEN CROSSINGS. 287 



If AxBi, A 2 B 2 , &c, are the pairs of collaterals, the leader has the greatest A, no 

 matter what be the coverticals. If several flaps have the greatest A, the leader 

 has the greatest B. If several have both A and B greatest, we compare co- 

 verticals. If one has the greatest C, it leads ; if more than one, the greatest D 

 gives the lead. If no leader can thus be determined, we have to examine the 

 collaterals of the A's. The leader has more than any other of the greatest of 

 these collaterals ; and so on we go over the collaterals of the B's, the C's, and 

 the D's. The leader, if there is only one. is certain to be found. 



I have never had occasion to examine the collaterals of AB,CD. If two 

 competitors have these all equal, it is almost a certainty that there is symmetry, 

 and no leader, but a set of co-leaders. Where there is no symmetry, no two 

 edges or flaps on a knot are alike. 



It suffices, after writing two or more flaps as equally claiming by their AB, 

 CD to lead, to place a note of interrogation, and to examine the symmetry, 

 which readily betrays itself. The deletion of any one of the co-leaders com- 

 pletes the reduction. 



A flap can neither be removed from a knot nor added to it without cutting 

 of threads and reunion of ends. But this does not trouble us here, as by art. 2 

 we know that every projection making tessaraces only is a true knot, that its 

 circles can be read by the rule of under and over, and that the threads of all 

 the circles can be drawn in double lines as narrow untwisted tapes visibly 

 passing under and over at alternate crossings. 



21. In the converse problem of construction, the question is, in how many 

 ways to add, on a knot P' of n — 2 crossings, a leading flap, so as to construct 

 without risk of repetition a subsolid of n crossings. The note of interrogation 

 written after the comparison of two flaps that can be drawn across two meshes 

 of P' is a presumption of symmetry, which is pretty certain to be verified when 

 we come to draw in turn our new flaps on P', and to examine the constructed 

 P of n crossings as to the leadership of the doubted flap which turns P' into P. 



22. Two things are to be noted here, both in reduction and construction. 

 If a flap /on any subsolid is covertical at its crossing a with a triangular mesh 

 abc, which carries a flap/ 7 on the edge be, since abc cannot lose a crossing by 

 the deletion of/ it thereby becomes itself a flap /" collateral with the flap/'. 

 Now a pair of collateral flaps is excluded from all our constructions, because it 

 is a circle of two crossings, whose projection represents nothing in space but a 

 movable ring through which one thread once passes. Wherefore the flap /is 

 indelible or a fixed flap. It cannot be removed, nor be a competitor for the 

 lead, either in reduction or construction. 



When two flaps are collateral with the same triangular mesh, both are fixed ; 

 for the deletion of either leaves the other hanging by a nugatory crossing which 

 admits a forbidden punctual section. The reader can easily verify this. 



