3 28 Proceedings of the Royal Society 
which would correspond to 
r 4 + 2 r 3 
for the right-handed part, and would give us the form 
or one of its deformations. 
The criterion by which to distinguish at once whether such 
symbolic representations as those just given represent knots or links 
is easy to find. If we remember that each of the (even number of) 
crossings lying on a closed curve is a corner of one black and of one 
white mesh (contained within the curve) — while each of the crossings 
lying within it is a corner of each of two white and of two black meshes 
— we see that unless we can enclose a part of the graphic symbol in 
such a way that the sum of the exponents within the enclosure, 
and that formed by the doubling of the number of the joining lines 
which are wholly within the enclosure, and adding it to the number 
of those which cut the boundary, are equal even numbers — the figure 
is necessarily a knot. But if we can enclose such a part, it requires 
to be farther examined to test whether the figure consists of links 
or is a single knot. 
Thus, in the example just given, the part 
/\ 
p — p 
\/ 
is a simple oval divided by two intersecting chords into three- 
cornered meshes — but in the following formula 
r 2 
Z\ 
/v* 3 
although the par 
seems to fulfil the conditions above, it does not represent a separ- 
— - 
