


166 PROFESSOR TAIT ON KNOTS. 
drawing the curve when the diagram is given is obvious. Thus the annexed 
diagram shows the result of the process as applied to a symmetrical symbol. 

An inspection of one of these diagrams shows at once 
(1.) The number of joining lines is the same as the number of crossings. 
Hence, as each line has two ends, the sum of the numbers representing the 
number of corners in either the black or the white spaces is twice the number 
of crossings. 
(2.) Every additional crossing involves one additional compartment, for the 
abolition of a crossing runs two compartments into one. But where there is 
no crossing there are two compartments, the inside and outside (Amplea, in 
Listine’s phraseology), of what must then be merely a closed oval. Thus 
when there are crossings there are +2 compartments. 
(3.) No compartment can have more than 7 corners. For, as the whole 
number of corners in the black or white compartments is only 2n, if one have 
more than n, the rest must together have less, and thus some of the joining 
lines in the diagram must wnite the large number to itself, 1.e., must give essen- 
tially nugatory intersections. 
As an illustration, let us use this process in giving a second enumeration or 
delineation of the forms of 7-fold knottiness. The numbering of the various 
forms is the same as that already employed in §§ 10, 11 above. 
5 vl 
=r | or —a3— 
N 2 | | 
- 
The second form of this symbol is particularly interesting as consisting of 
two parts. This accords with the composite nature of the knot. 
3— 3 : 
=) 3 l | WSs 
3 
NI \ , NI 
IT. III. IV. 
