
168 PROFESSOR TAIT ON KNOTS. 
§ 21. A tentative method of drawing all possible systems of closed curves 
with a given number (7) of double points is thus at once obvious. 
Write all the partitions of 27, in which no one shall be greater than m and 
no one less than 2. Join each of these sets of numbers into a group, so that 
each number has as many lines terminating in it as it contains units. Then 
join the middle points of these lines (which must not intersect one another) by 
a continuous line which zntersects itself at these middle points and there only. 
When this can be done we have the projection of a knot. When more 
continuous lines than one are required we have the projection of a linkage. 
To give simple examples of this process, let us limit ourselves to 4 and 5 
intersections. 
The only partitions of 8, subject to the conditions above, are 
(1):4 4 
(2)4 2 2 
(Qyerge 2 
(4)2 2 2 2 
Now the number of black and white compartments together must in this case 
be 4+2. Hence there are but four combinations to try, viz., (1) and (4), (2) 
and (2), (3) and (3), (2) and (3). Of these, the last is impossible ; the others are 
as in Plate XVI. fig. 16. The third is the amphicheiral knot already spoken 
of, and the second may for the same reason be called an amphicheiral link. 
The partitions of 10, subject to our rule, are 
5 
oP PR BR oT 
oo dD Oo HB OD 
22 
and the four figures (Plate XVI. fig. 17) give the only valid combinations of 
these. The third and the first are the knots already described (§ 18), the 
others are links. 
§ 22. The spherical projection already mentioned (§ 15) will in general allow 
us to regard and exhibit any knot as a more or less perfect plait. It does so 
perfectly whenever the coil is clear, 7.e., when all the windings of the cord may be 
regarded as passing in the same direction round a common vertical axis thrust 
through the knot. When the coil is not clear some of the cords of the plait are 
doubled back on themselves. Thus by drawing the plait corresponding to a 
given scheme we can tell at once whether one of its forms is a clear coil or not. 
