WITH A CENSUS OF THE AMPHICHEIRALS WITH TWELVE CROSSINGS. 237 
compartment symbol. Little mentions the fact that the knot is not uniquely 
determined unless its so-called partition symbol, which is the same as the compart- 
ment symbol, as defined above, indicates the sequence of- the crossings on the knot. 
The compartment symbol of the knot (fig. l) is 
where the first refers to the set of S compartments and the second to the set of 
A compartments. In the following pages these symbols are referred to as primary 
and secondary. As Tait showed, the knot may be constructed from either the 
primary or secondary symbol by connecting the mid-points of the joins with a line 
which intersects itself only at these points. 
§ 2. Methods of Varying a Given Knot. 
Since the positions of the apparent double points of a twisted wire in space vary 
as the point of view changes, the plane projection is susceptible of two types of 
variation, which Tait calls deformation and distortion respectively. 
By deformation of the knot Tait means a projection which leaves unaltered the 
relative positions of the compartments, as well as the number of angles in each 
compartment. For instance, any compartment may be made the amplexus, or 
infinite compartment, by turning the knot into this compartment ; or, which is 
precisely equivalent, by inversion of the knot with respect to a point of this com- 
partment as origin. But the knot scheme is unaltered by deformation, and the two 
knots are said to be equivalent. 
Distortion, on the other hand, is a projection which changes the position of one 
or more of the crossings, so that, in general, it is impossible to represent the distorted 
form by the same scheme. In certain cases the number of angles and the arrange- 
ment of the joins of the various compartments is unaltered by a distortion. Such a 
distortion, therefore, reduces to a deformation, and the distorted form is equivalent 
to the original. As regards the knot in the plane, distortion is the process of shifting 
a crossing from one lap of the thread to another by a twist through two right angles 
of a limited portion of the knot. For example, by a rotation through two right 
angles about an axis in the plane of the paper, downwards through the crossing a, 
fig. 2 is distorted into fig. 2'. 
Effect of Distortion on the Alphabetical Symbol . — If from a limited portion of a 
knot there emerge two free ends, that is to say, if a single part of the complete 
thread exhibits a certain number of crossings, then the knot consists of two or more 
separate knots on the one thread, and is said to be composite. 
