288 REV. T. P. KIRKMAN ON THE ENUMERATION, DESCRIPTION, AND 



By continuing this reduction of a subsolid by removal of the leading fl ip, we 

 must at last arrive either at a solid knot, or at one of the two irreducibles 3 A 

 and 4 A, of three and of four crossings. 



23. Construction of Knots o/n Crossings. — The rules for this are the exact 

 converse of those above given for reduction. 



First, to construct the subsolids of n crossings, all inferior knots being given 

 with their symmetry, we have in the first place to take in turn every subsolid 

 P' of » — 2 crossings, and to determine before we draw them the different lead- 

 ing flaps that can be added on P'. Knowing its symmetry, we can write down 

 and mark on its edges every different pair of points on flap or edge that can be 

 joined as the crossings of a new flap, and also the collaterals and coverticals 

 which this will have. We make a table of the possible leading flaps, with the 

 notes of interrogation that presume symmetry in the P of n crossings to be 

 built on P'. Next we draw the leading flaps, thus constructing and registering 

 the resulting subsolids P. 



A caution is required here, for the examination of the claim of a new flap, 

 ab = 3M, to leadership ; a and b being the crossings of the flap, when one of its 

 collaterals is a triangle abc. If c in this triangle is the crossing of a flap cd, cd 

 becomes fixed (art. 22), for it is covertical with a triangle^aJ, which carries a 

 flap on its edge ab. Care must be taken to exclude this flap cd from claim to 

 leadership over ab. 



I was caught in this trap in the art. 41, for I had entered the flap (bd) as 

 led by (56), and thus missed the unifilar 8 Gr. Professor Tait found this knot, 

 adding one to my first list of 8-fold knots. He first found also the unifilars 

 9 A/, gAk, and 9 B^, omitted by a like error in arts. 51 and 56. He also first 

 found Dm, which I ought to have constructed along with 9 DZ in art. 61. 



24. In the second place, we take in turn every unsolid P" of n — 2 crossings 

 on which a leading flap or flaps can be drawn so as to abolish all concurrences, 

 and to block linear section. Such leading flaps will be few. Next we draw 

 them all, and thus complete without omission or repetition our list of subsolids 

 P of n crossings. This list is the only difficulty of our work ; what follows is 

 for every value of n all easy routine, as we shall see ; but it soon becomes too 

 tedious by the enormous number of results to be registered and figured. 



25. Next, to construct the unsolids of n crossings which have no concurrence, 

 we impose in the first place on the solids and subsolids, and in the second on 

 the unsolids, of n — i crossings, each taken in turn as the subject Q' to be 

 charged, e charges of least marginal subsolids, all of k crossings, no matter 

 what be the number of their meshes, so as to add to Q' ek = i crossings, com- 

 pleting an unsolid Q of n crossings without a concurrence. 



The e charges may be all or none alike, or all but one alike, &c. ; and from 

 our list of subsolids of k crossings must be selected with or without repetition 



