
PROFESSOR TAIT ON KNOTS. 147 ° 
over and under. It has been pointed out to me that this seems to have been 
long known, if we may judge from the ornaments on various Celtic sculptured 
stones, &c. It was probably suggested by the processes of weaving or plaiting. 
I am indebted to Mr Datuas for a photograph of a remarkable engraving by 
Durer, exhibiting a very complex but symmetrical linkage, in which this alter- 
nation is maintained throughout. Formal proofs of the truth of this and some 
associated properties of knots will be found in the little paper already referred 
to.* They are direct consequences of the obvious fact that two closed curves 
in one plane necessarily intersect one another an even number of times. It 
follows as an immediate deduction from this that in going continuously round 
any closed plane curve whatever, an even number of intersections is always 
passed on the way from any one intersection to the same again. Hence, of 
course, if we agree to make a knot of it, and take the crossings (which now 
correspond to the intersections) over and under alternately, when we come back 
to any particular crossing we shall have to go wader if we previously went over’, 
and vice versd. This is virtually the foundation of all that follows. 
But it is essential to remark that we have thus two alternatives for the cross- 
ing with which we start. We may make the branch we begin with cross wader 
instead of over the other at that crossing. This has the effect of changing any 
given knot into its own image in a plane mirror—what Listine calls Perversion. 
Unless the form be an Amphicheiral one (a term which will be explained later), 
| this perversion makes an essential difference in its character—makes it, in fact, 
a different knot, incapable of being deformed into its original shape. 
LisTInG speaks of: crossings as dexiotrop or laeotrop. If we think of the 
edges of a flat tape or india-rubber band twisted about its mesial line, we 
recognise at once the difference between a right and a left handed crossing. 
(Plate XV. fig. 1.) Thus the acute angles in the following figure are left 
handed, the obtuse, right banded ; and they retain these characters if the figure 
be turned over (z.¢., about an axis in the plane of the paper) :— 
PS 
but in its image in a plane mirror these characters are interchanged. 
§ 2. Suppose now a knot of any form whatever to be projected as a shadow 
cast by a luminous point on a plane. The projection will always necessarily 
have double points,t and in general the number of these may be increased— 
* “Messenger of Mathematics,’ January 1877. 
t Higher multiple points may, of course, occur, but an infinitesimal change of position of the lumin- 
ous point, or of the relative dimensions of the coils of the knot, will remove these by splitting them 
into a number of double points, so that we need not consider them. 
