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XXVI. — The 364 Unifilar Knots of Ten Crossings, Enumerated and Described. 

 By Rev. Thomas P. Kiekman, M.A., F.R.S. 



(Read July 20, 1885.) 



1. The 119 subsolids (marked ss) and the 244 unsolids (marked us), of these 

 unifilars are here arranged in lists according to their flaps. F e is the num- 

 ber of flaps of e loops upon a knot ; and the headings of the lists, as, e.g. 10 I, 

 F 2 = l, F 1 = 3, describe so far all the knots in the lists. Thus in 10 I each has 

 one 2-ple flap and three single ones. After the number in the list comes always 

 the base on which the knot is constructed by the rules of my paper, XVII. in 

 vol. xxxii. part ii. of the Trans. R. S. E. ; and the reader who has that paper 

 before him will find it easy to draw any knot on its base, nearly always there 

 figured, by the first given flap, which is the leading one of the knot described ; 

 thus in 10 I the first written flap is the double one, and in 10 P, F„ — 1, F x = 2, it 

 is the triple one. The leading flap of a subsolid is always followed by a colon. 



It will be seen that no two subsolids nor unsolids have the same description. 

 A flap AB, CD is generally given by its collaterals AB only ; but the coverticals 

 CD are added when required for distinguishing the knots from each other. 



To me it appears that this tabulation of these knots will be more useful 

 than the engraving of the figures; for the student who draws a 10-fold unifilar 

 will hereby more readily satisfy himself that it is found or not found in my 

 census, than if he had 364 knots projected before him, in the manner of the 

 plates of my former paper. One solid knot makes up 364 unifilars. 



2. I am indebted to Professor Tait for the detection of several bifilars which 

 I had passed as unifilars, and for the addition of four to my list of unifilars, — 

 namely, 10 B,21 ; ]0 D,17 ; 10 I,30 ; and ]0 L,8 ; and I may obtain from him farther 

 contributions before he has performed on the figures all his surprising feats of 

 twisting, which add a charm of conjuring to this curious and difficult inquiry. 



The abbreviations which mark the symmetry are those used and explained 

 in my paper above mentioned. My linear drawings of these unifilars of ten, as 

 well as the more numerous figures of the unifilars of eleven crossings, will be 

 found in the archives of the Royal Society of Edinburgh. 



I have to acknowledge two omissions in my census of the knots of nine 

 crossings. One is that of the bifilar 9 Az 2 referred to as a base under 10 I. This 

 ought to have been formed in art. 55 by drawing from the point c the flap 63,44. 



The other knot omitted is a unifilar ( 9 Ar 2 ) which should have been formed 



VOL. XXXII. PART III. 4 K 



