CONSTRUCTION OF KNOTS OF FEWER THAN TEN CROSSINGS. 285 



When a concurrence of n stands about the (n + k + 2)-gonal mesh F, their 

 common collateral, F is reduced by deletion of n — 1 flaps, each conceived to 

 vanish by the union of its two summits, to the {k + 3)-gon F', carrying one only 

 of those flaps. 



If their collateral F is an (n + l)-gon, it is reduced by the vanishing of n — 2 

 of the concurrent flaps to a triangle/ carrying the remaining two. This/ cannot 

 lose a flap without losing an edge and disappearing. 



Every concurrence on the unsolid Q of art. 12, whether standing on a 

 marginal or non-marginal component of Q, is to be thus reduced, and Q now 

 becomes an unsolid Q' without a concurrence, of n — i crossings. 



14. Least Marginal Sections and Least Marginal Subsolids. — Our unsolid Q', 

 obtained by deletion of i flaps, has one or more linear sections, marginal or not. 



By a marginal linear section of Q' can be cut away one and only one 

 marginal subsolid, on which (art. 9) lies no linear section except through the 

 two crossings of a flap. 



By a least marginal section of Q' can be cut away a. least marginal subsolid. 



A marginal subsolid of k crossings all untouched by any possible least 

 section, is a least marginal, when no marginal subsolid of fewer than k cross- 

 ings untouched by the section can be cut away from Q' by any kind of section. 



15. The Five Kinds of Linear Section of an Unsolid Knot. — These are, 



ffc, which is read, flap on flap close ; 

 /, „ „ flap on flap ; 

 fe, „ „ flap on edge; 

 ef, „ „ edge on flap ; 

 ee, ,. „ edge on edge. 



The first letter in these symbols denotes the flap or edge of the least 

 marginal subsolid cut away from, or in construction imposed on, the unsolid or 

 subsolid charged. 



16. The linear section ffc is the only one that passes through two crossings. 

 After making the section ffc, there is a pair of truncated crossings, both on the 

 diminished Q' and on the subsolid removed. These are completed into 

 tessaraces in the severed portions by replacing each portion on the other by the 

 two pairs of edges of two flaps, at the cost of which two flaps the portions were, 

 in construction at the section ffc, united to form Q'. 



These pairs are conceived to be so united to the broken threads at the 

 truncated crossings as to complete both the severed portions into two knots. 

 This can always be done, and needs not trouble us, when we are reducing a 

 projected figure of simple lines making tessaraces only. 



If the subsolid S removed at this section ffc has, when completed by its 

 restored flap, k crossings, the n — i crossings of the undiminished Q' are made 

 fewer by k—2 ; for Q' has lost only the crossings of S not on the section. 



