248 
MARY GERTRUDE HASEMAN : ON KNOTS, 
circle in a point, the inverse curve will touch it at the diametrically opposite point, 
and it is necessary only to replace the contacts by crossings to secure the repre- 
sentation of a unifilar knot which exhibits amphicheiralism. 
The second method of producing compartments S- f and \ of the desired nature 
is rejected by Tait, since it leads to a link solution. While a closed curve and its 
image in a diametral plane, together with the great circle of the sphere in this 
plane, will give the desired arrangement of compartments, it is impossible to fuse 
the curves into a single circuit as in the first case, for the process introduces triple 
points which cannot be replaced by three dps without destroying the amphicheiral 
symmetry. Hence, only the simplest trifilar link can result from such an arrange- 
ment. If the curve is taken as its own image, a bifilar knot is represented. 
Before applying this method of construction, some preliminary considerations 
are necessary. 
A closed curve on a sphere is either* a simple circuit or one member of a twin 
circuit. The simple circuit, which is its own opposite on the sphere, is met by a 
great circle in an odd number of pairs of points. In Tait’s method of construction 
a simple circuit leads always to an m-filar knot. 
The twin circuit, which consists of a closed curve and its opposite on the sphere, 
intersects a great circle in an even number, 2*, of pairs of points. There are two 
types of twin circuits which may present themselves. First, if |||0, each member 
of the twin circuit is confined to a single hemisphere. Second, if l =/=0, each member 
exceeds that hemisphere, and therefore the two members may intersect, necessarily 
in an even number, 2&' of points where o-' = 0, 1, 2, . . . 
The projection,/, of any closed plane curve on to the sphere from its centre gives 
a twin circuit of the first type. A twin circuit of the second type is obtained by 
a similar projection of a plane curve of even order, which cannot be projected 
entirely into the finite part of the plane, as, for example, the Cayley non-singular 
sextic,t for which ' = 6, </=0. Every non-singular twin circuit divides the sphere 
into three regions, in one of which an odd circuit may lie. 
Each member of the twin circuit may have <r dps, thus giving rise to & pairs of 
crossings in the resulting knot. The only 'other cause that can produce crossings 
in the knot is the presence of k pairs of contacts of the circle and the twin circuit. 
For a knot of order n, k 
may not be greater than 
n 
2 ' 
Hence for w= 12, k F 6. 
The few numerical possibilities for the above numbers to be considered are given 
in the following table : — 
k 111222233344456 
t 4224220200 2 0000 
xr 131002413102010 
<r' 002020000200200 
* Mobius, Tiber die Grundformen der Linien der dritten Ordnung, ii, p. 90. 
f Cayley, vol. v, op. 361, p. 468. 
