Prof. Tait on Listing's Topologie. 43 



All of these remarks will be obvious from any one of the 

 three (equivalent) diagrams 9, 10, or 11. 



(18) As I have already said, the subject of knots affords 

 one of the most typical applications of our science. I had 

 been working at it for some time, in consequence of Thom- 

 son's admirable idea of Vortex-atoms, before Clerk-Maxwell 

 referred me to Listing's Essay; and I had made out for 

 myself, though by methods entirely different from those of 

 Listing, all but one of his published results. Listing's re- 

 marks on this fascinating branch of the subject are, unfortu- 

 nately, very brief ; and it is here especially, I hope, that we 

 shall learn much from his posthumous papers. In the Vor- 

 studien he looks upon knots simply from the point of view of 

 screwing or winding ; and he designates the angles at a 

 crossing of two laps of the cord by the use of his A and 8 

 notation (§ 4). Fig. 12 will show the nature of such cross- 

 ings. Figs. 13, 14, and 15 show what he calls reducible and 

 reduced knots. In a reducible knot the angles in some com- 

 partments at least are not all X or all 8 (the converse is not 

 necessarily true). In a reduced knot, each compartment is 

 all A, or all S. 



(19) My first object was to classify the simpler forms of 

 knots, so as to find to what degree of complexity of knotting 

 we should have to go to obtain a special form of knotted 

 vortex for each of the known elements. Hence it was neces- 

 sary to devise a mode of notation, by means of which any 

 knot could be so fully described that it might, from the de- 

 scription alone, be distinguished from all others, and (if 

 requisite) constructed in cord or wire. 



This I obtained, in a manner equally simple and sufficient, 

 from the theorem which follows, and which (to judge from 

 sculptured stones, engraved arabesques, &c.) must have been 

 at least practically known for very many centuries. 



Any closed plane curve, which has double points only, may 

 be looked upon as the projection of a knot in ivhich each portion 

 of the cord passes alternately under and over the successive laps 

 it meets. [The same is easily seen to hold for any number of 

 self-intersecting, and mutually intersecting, closed plane 

 curves, in which cases we have in general both linking and 

 locking in addition to knotting.] 



The proof is excessively simple (§ 11). If both ends of 

 one continuous line lie on the same side of a second line, there 

 must be an even number of crossings. 



(20) To apply it, go continuously round the projection of 

 a knot (fig. 16), putting A, B, 0, &c. at the first, third, fifth, 

 &c. crossing you pass, until you have put letters to all. 



