36 Proceedings of Royal Society of Edinburgh. [dec. 20 , 
2. Note on Knots on Endless Cords. By A. B. Kempe, Esq. 
Communicated by Prof. Tait. (Plate I.) 
1. Each crossing divides the cord into two loops. 
2. Any other crossing either lies on both these loops, say is 
linked to the former crossing, or on one loop only, say is not linked 
to the former crossing. 
3. It can be shown without difficulty that if crossing a is linked 
to crossing b, then crossing b is linked to crossing a ; and therefore, 
also, if crossing a is not linked to crossing b, crossing b is not linked 
to crossing a. 
4. Hence pairs of crossings are of two sorts, viz., linked and 
unlinked. 
5. We have the two following fundamental laws : — 
(a) The number of crossings linked to each crossing is even. 
( b ) If two crossings are not linked to each other, the number 
of crossings linked to both is even. 
6. We may represent a knot diagrammatically thus : — 
Represent the crossings by small circles or nuclei. 
Join pairs of nuclei which represent pairs of linked crossings by 
lines or links. 
No lines are to be drawn in the case of unlinked pairs of crossings. 
7. In these diagrams, in conformity with sec. 5, the number of 
links proceeding from a nucleus must be even, and if two nuclei are 
not joined by a link, the number of nuclei joined by links to both 
must be even. 
8. This mode of representing knots has the advantage of indicating 
the degree of complexity of the various knots. Thus, the diagram 
representing two distinct knots on the same string will consist of 
two entirely detached portions, and nugatory crossings will be re- 
presented by nuclei having no links proceeding from them. 
9. In the plate the various knots of three , four , Jive, six , and 
seven crossings are indicated in outline on this plan. 
10. It will occasionally be convenient to represent pairs of cross- 
ings which are unlinked by pairs of nuclei joined by a dotted line 
or link, pairs of crossings which are linked not being joined at all. 
