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XVII. — The Enumeration, Description, and Construction of Knots of Fewer 

 than Ten Crossings. By Kev. Thomas P. Kirkman, M.A., F.R.S. (Plates 

 XL.-XLIII.) 



(Eead June 2, 1884) 



1. By a knot of n crossings, I understand a reticulation of any number of 

 meshes of two or more edges, whose summits, all tessaraces (a/07), are each a 

 single crossing, as when you cross your forefingers straight or slightly curved, 

 so as not to link them, and such meshes that every thread is either seen, when 

 the projection of the knot with its n crossings and no more is drawn in double 

 lines, or conceived by the reader of its course when drawn in single line, to 

 pass alternately under and over the threads to which it comes at successive 

 crossings. 



The rule for reading such a reticulation of single lines meeting in tessaraces 

 only is this — Coining by the edge or thread pq to the tessarace q, you leave it 

 by the edge of q which makes no angle with pq, nor is part of thread under or 

 over which you pass at q. 



2. It is not necessary, after what Professor Tait has written on knots, to 

 prove that every reticulation having only tessarace summits, whether polyedron* 

 or not, if it be one continuous figure and projected to show all its crossings and 

 no more, can be read all through by this alternate under and over, so that all 

 its closed circles, one or more, can be written down in numbered summits, and 

 that the knot can be labelled as unifilar, bifilar, or trifilar, &c. 



3. If a thread a of a knot, after passing under or over a thread b, passes 

 over or under b before it meets a third thread c, there is a linkage of two cross- 

 ings and a flap between them. This flap is the small eyelet seen between two 

 links of a slack chain as it lies on the table : it is a 2-gon, a mesh of two edges 

 and of two crossings. Here see art. 9. 



4. In our enumeration of knots of n crossings, two, C and C, are counted 

 as the same one, whenever and only when, in the number, polygonal rank, and 

 order of their meshes C is either the exact repetition or the mirrored 

 image of C; and I consider the threads of all the circles of a knot to be tape 

 untwisted. 



5. Nothing general seems to have been written upon knots of more than 

 seven crossings. Nor, fortunately for the claims of those knots upon the inte- 



* I hope to be pardoned for omitting the h. It annoys me to hear the learned say polyhedron. 

 Why not periliodic also 1 or, more learnedly, perihodic ? 



VOL. XXXII. PART II. 3 A 



