44 Prof. Tait on Listing's Topologie. 



Then go round again, writing down the name of each crossing 

 in the order in which you reach it. The list will consist of 

 each letter employed, taken twice over. A, B, C, &c. will 

 occupy, in order, the first, third, fifth, &c. places ; but the 

 iv ay in which these letters occur in the even places fully charac- 

 terizes the drawing of the projected knot. It may therefore 

 be described by the order of the letters in the even places 

 alone ; and it does not seem possible that any briefer descrip- 

 tion could be given. 



To prove that this description is complete, so far as the 

 projection is concerned, all that is required is to show that 

 from it we can at once construct the diagram. Thus let it 

 be, as in fig. 16, E F B A C D. Then the full statement is 

 AEBFCBDAECFD/A&c. 



(21) To draw from such a statement, choose in it tw T o 

 apparitions of the same letter, between which no other letter 

 appears twice. Thus AECFD/A (at the end of the state- 

 ment) forms such a group. It must form a loop of the curve. 

 Draw such a loop, putting A at the point where the ends 

 cross, and the other letters in order (either way) round the 

 loop. Proceed to fill in the rest of the cycle in the same 

 way. The figures thus obtained may present very different 

 appearances ; but they are all projections of the same definite 

 knot. The only further information we require for its full 

 construction is whicli branch passes over the other at each par- 

 ticular crossing. This can be at once supplied by a + or — 

 sign attached to each letter where it occurs in the statement 

 of the order in the even places. 



(22) Furnished with this process, we find that it becomes 

 a mere question of skilled labour to draw all the possible 

 knots having any assigned number of crossings. The re- 

 quisite labour increases with extreme rapidity as the number 

 of crossings is increased. For we must take every possible 

 arrangement of the letters in the even places, and try whether 

 it is compatible with the properties of a self-intersecting plane 

 curve. Simple rules for rejecting useless or impracticable 

 combinations are easily formed. But then we have again to 

 go through the list of survivors, and reject all but one of each 

 of the numerous groups of different distortions of one and the 

 same species of knot. 



1 have not boon able to find time to carry out this process 



further than the knots with seven crossings. But it is very 



remarkable that, so far as I have gone, the number of knots 



of each class belongs to the series of powers of 2. Thus : 



Number of crossings . . 3, 4, 5, 6, 7, 



Number of distinct forms 1, 1, 2, 4 ; 8. 



