246 
MARY GERTRUDE HASEMAN : ON KNOTS, 
laps of the thread, and consequently any path drawn across the knot can be made 
a great circle. 
Since the two parts of the knot are congruent, rotation about a certain diameter 
will bring the first part of the knot into the position originally occupied by the 
second part. Tait shows that this diameter must terminate in the mid-points of 
corresponding laps of the thread. But a rotation about such a diameter necessitates 
the existence of two pairs of adjacent corresponding compartments, in order that each 
compartment of the sphere may be rotated into the position of its corresponding 
compartment. The number of such diameters depends on the number of pairs of 
adjacent corresponding compartments. Now deform the knot so that the path from 
0 to O', which meets the knot in the minimum number, p , of points exclusive of 
0, O', shall become the arc of a great circle S of the sphere, but in such a way as to 
keep corresponding crossings at equal arcual distances from the points 0, O'. Since 
all great circles through the points 0, O' divide the knot into congruent halves, the 
projection of the knot from the point O' on the tangent plane to the sphere at 0 
will be divided into halves by all the straight lines through 0, and in particular by 
the straight line s, which corresponds to the great circle S of the sphere. Of the 
2p + 2 points of intersection of the line s and the knot, one, corresponding to the 
point O', lies at infinity, and the rest by pairs at equal distances from the amphicheiral 
centre 0. A part of the thread which joins two of these points is a bend. The 
framework for one half of the knot, that is, the framework on either side of the line s, 
consists of p + 1 bends, of which one is infinite, since one point of intersection lies at 
infinity. Every possible arrangement of the bends must be considered ; and in every 
admissible arrangement the bends are made to intersect so as to exhibit one half of 
the total number of crossings. The congruent half completes the knot (cf fig. 5). 
Inasmuch as two entirely different paths, 0, O', through the knot may give the 
proper number of intersections, 2p + 2, the knot # maybe built upon an entirely 
different framework, and in such a case the eye may be deceived. The equivalence of 
the two is immediately detected by means of the compartment or intrinsic symbol. 
If the knot is projected from the point 0, on the tangent plane at the point O', 
the figure exhibits symmetry about the point O'. This projection may be said to be 
* Tait, Trans. Roy. Soc. Edin., xxxii, p. 496 ; or Scientific Papers, i, p. 338. 
