494 PROFESSOR TAIT ON KNOTS. 



In § 17 of my first paper I introduced it thus : — 



An amphicheiral knot is one which can be deformed into its own perversion. 



The word " deformed" was here used in the sense of alteration of form by- 

 mere change of point of view, or mode of projection; a process which leaves 

 the number of corners in each mesh, and the relative positions of the various 

 meshes, unchanged. This definition implies that the right and left handed 

 meshes are similar in pairs and similarly situated in congruent groups ; and 

 it will be adhered to for the present, though we shall afterwards find that there 

 are at least three other senses in which a knot may be called amphicheiral, and 

 shall thus be led to speak of different orders and classes of amphicheirals. The 

 above definition will then be considered to belong to amphicheirals of the First 

 Order and First Class. 



2. Suppose an amphicheiral knot to be constructed in cord, and extended 

 oyer the surface of a sphere which swells out when necessary so as to keep the 

 cord tight like the netting on a gazogene. Let its various laps be displaced 

 until the several corresponding pairs of right and left handed meshes are made 

 equal as well as similar. Trace its position on the sphere. Now suppose it 

 to become rigid, and move it about on the surface of the sphere. We can 

 again bring it to coincide with its former trace, but in such a way that each 

 left-handed compartment now stands where the corresponding right-handed 

 one was, and each right-handed where its corresponding left-handed was. 

 Now such a displacement, as we know, can always be effected by a finite 

 rotation about a diameter of the sphere as axis. 



This axis, of course, cannot terminate (at either end) inside a mesh, else 

 that compartment could not be shifted by the rotation to the original position 

 of the corresponding one of the other kind. Hence either end of the axis 

 must be at a crossing, or midway on the lap of cord passing through two 

 adjoining crossings. A little consideration shows that if one end be at a 

 crossing the other also must be at a crossing, and the whole must be a link. 

 This is easily seen from the fact that, if one end of the axis be at a crossing, the 

 four meshes which meet there must each exactly fit that next it when the 

 whole is turned through a right angle ; and the series which immediately 

 surrounds these must possess a similar property, &c, &c. Thus the whole 

 spherical surface must be covered with a pattern which consists of four equal 

 and similar parts, each of which takes the place of the preceding one at every 

 quarter of a rotation about the axis. And four laps of the string must there- 

 fore proceed all in the same way from one end of the axis to the other; 

 since, if we can trace one lap of the string continuously from one crossing to 

 the other, exactly the same must be true of the other three. [Of course, if the 

 string cannot be traced from one crossing to the other, there must be two 

 separate strings at least.] 



