244 
MARY GERTRUDE HASEMAN: ON KNOTS. 
If such a knot is fitted on the surface of a sphere so that the corresponding 
arcual boundaries are made equal, a spherical compartment ^ and its corresponding 
compartment \ must either be congruent or symmetrical * (where symmetrical 
is used in the sense of symmetrical triangles on a sphere). An amphicheiral knot 
whose corresponding compartments are congruent is called by Tait an amphicheiral 
of the first order, as distinguished from one of the second order, in which the 
corresponding compartments are merely symmetrical. 
From the definition of an amphicheiral knot it is seen that every distortion D ra 
carries with it a conjugate distortion D n such that the product D,J) n — that is to 
say, the simultaneous application of the two distortions— gives an amphicheiral 
knot. The form obtained by the single operation D n can be distorted into its own 
perversion by the operator D^D^, and is said to be of the second class, while one 
Fig. 4". 
Fig. 4"'. 
which can be deformed into its own perversion belongs to the first class. Therefore 
Tait divides the amphicheirals of each order into those of the first or second class, 
according as they are the result of operating on the knot with conjugate or 
non-conjugate distortions. 
In an investigation of the amphicheiral knots of order 12 it appears that a 
third classification of amphicheiral knots of the first and second orders is necessary, 
namely, amphicheirals which are obtained as the product of two or more non- 
conjugate distortions. For example, consider the amphicheiral knot (fig. 4) whose 
compartment symbol is 
* {Trans. B.S.E., xxxii, p. 494 ; or Scientific Papers , i, p. 336.) In his third paper Tait deliberately limits him- 
self to this view ; but he remarks — “ We shall afterwards find that there are at least three other senses in which a 
knot may be called amphicheiral, and shall thus be led to speak of different orders and classes of amphicheirals.” 
(See below, § 6.) 
