28 C. N. Little 
Knots, with a Census for Order Ten. 
no ambiguity we shall also call the number representing a compart- 
ment a part, and either group of compartments a partition. 
Since every closed plane curve of 7 crossings, having double points 
only, may be read alternately over and under at the crossings, every 
such curve which gives parts, none greater than n or less than 2, 
may be taken as a projection of a reduced knot of » crossings. We 
call such curves knot-forms or briefly, forms, and regard two forms 
as distinct if they do not have the same parts similarly arranged. 
The first part of the problem is to find all the different knot-forms 
of any order. 
Since the same knot may be transformed so as to be projected into 
more than one knot-form, the second part of the problem is, from the 
complete series of knot-forms of » crossings to find all the different 
n-fold knots. Knots exist for which the law of over and under does 
not hold; these are not considered in the present paper. 
3. It is unnecessary to do more than allude to two very distinct and 
very ingenious methods devised and used, the one by Tait and the 
other by Kirkman, for the solution of the first part of this problem. 
We may perhaps infer from Professor Tait’s opinion* that “a full 
study of 10-fold and 11-fold knottiness seems to be relegated to the 
somewhat distant future,” that they were more laborious than proves 
to be necessary. 
4, A third method, based on Listing’s type-symbol, is thus de- 
scribed by Professor Tait at page 168 of his first memoir. 
“Write all the partitions of 22, in which no one shall be greater 
than » and no one less than 2. Join each of these sets of numbers 
into a group, so that each number has as many lines terminating in 
it as it contains units. Then join the middle points of these lines 
(which must not intersect one another), by a continuous line which 
intersects itself at these middle points and there only. When this can 
be done we have the projection of a knot. When more continuous 
lines than one are required we have the projection of a linkage.” 
On page 160 of the same memoir, he says, speaking of this 
method: ‘‘ But we can never be quite sure that we get all possible 
results by a semi-tentative process of this kind. And we have to try 
an immensely greater number of partitions than there are knots, as 
the great majority give links of greater or less complexity.” 
It seems possible however, with the help of some simple theorems 
to make the “ Partition Method ” exhaustive, and wholly to do away 
with the drawing of links. 
* In 1884. Trans. R. 8S. E., xxxii, 328 
EE 
