236 
MARY GERTRUDE HASEMAN : ON KNOTS, 
compartments by the projection of the knot of order n. Both Listing and Tait 
showed that, of these compartments, no one contained less than 2 or more than n 
angles. Following Listing’s notation, the angle on the left along with its vertical, 
as a crossing is passed by the upper thread, is denoted by 8 and the remaining pair 
by A. The various compartments of an alternating knot are monotype ; that is to say, 
the angles are of the same character, as shown in fig. 1. The Listing type-symbol 
is merely an enumeration of the two sets of compartments, in which an exponent 
is used to indicate the number of angles in a compartment and a coefficient to 
represent the number of such compartments. Thus the Listing type-symbol for the 
knot given in fig. 1 is 
28 2 + 2S 3 + 8 4 + 28 5 
2A 2 + 2 A. 3 + 2A 4 + A 6 
In general^ each part of the Listing type-symbol for a knot of order n amounts to 
nothing more than a set of partitions of the number 2 n, where each member of the 
partition indicates the presence of a compartment with the same number of angles as 
there are units in this member. For example, the partitions corresponding to the 
type-symbol above are : 
2 2 3 3 4 5 5 
2 2 3 3 4 4 6 
But it is not sufficient for the determination of the knot to know simply the 
number of compartments and the number of angles in each. It is necessary to know 
the number of joins between the various compartments as well as the arrangement 
of these joins. The number and arrangement of these joins is expressed by the 
compartment symbol, in which the joining lines indicate the number as well as the 
arrangement of the crossings connecting the set of 8 compartments and the number 
as well as the arrangement of the laps of the thread bounding the set of A compart- 
ments, or vice versa. We may assume that Tait recognised the importance of the 
order of the crossings, for his symbols conform to the above definition of the 
