32 C. N. Little—Knots, with a Census for Order Ten. 
joining them so as to preserve the law of over and under one string 
will be formed. While if from the 27th crossing the strings are car- 
ried around, that string over at the Ist is under at the 27th and, on 
joining, there will be two strings. 
Hence, as is well known, ue is always a knot, while yes 
always a link. 
12. For the purpose of distinguishing between clutches giving 
knots and those giving links, it follows from Theorem III that we 
“may take the direct bonds between any two parts together, and these 
form open coils of the subordinate partition ; and further it is evident 
from § 11 that any odd open coil (even number of bonds), may be 
dropped, and any even coil (odd number of bonds) may be replaced 
by a single bond. If the resulting clutch gives link, so would the 
original. If the resulting clutch be still too complex for easy recog- 
nition of its character, the clutch of the subordinate partition of the 
resulting form may perhaps be still farther reduced in the same way. 
If the clutches of lower orders were at hand they also could be used 
for settling the question. 
13. We have the following theorems for throwing out clutches 
unproductive of knot forms. 
IV. Turorem.—lIf a part be joined to other parts in every case by 
an even number of bonds, there is linkage. For, the string about 
this part is closed by Section 12. 
V. THrorEM.—If two parts are connected by two 2-gons (of the 
same partition with the parts) there is linking. For they may be put 
in succession by Theorem III. When this is done there is a 4-gon of 
the negative partition bound to two other parts in each case by two 
bonds, and IV applies. 
VI. Tarorrm.—An odd part joined to one part by an odd number 
of bonds and to other parts in every case by an even number of 
bonds may be dropped; for, by Theorem III it becomes a loop with a 
single crossing, and this can have no effect on the question of linking. 
In particular a 3-gon joined by one bond to one part and by two 
bonds to a second part may be dropped. 
If two odd parts are joined by an odd number of bonds, and are 
joined to other parts in every case by an even number of bonds there 
is linkage. 
In particular two 3-gons so joined throw out the clutch. 
VIL. Tuzorem.—lIf two 3-gons, C and D, are themselves joined 
directly and are joined to A and B in each case by a single bond 
there is linkage. 
