PROFESSOR TAIT ON KNOTS. 499 



curves traced by the ends of the diameter must intersect one another, which 

 implies that they must each cut the great circle in two points at least besides 

 touching it at one or more. Hence the lowest knottiness in which they can 

 possibly occur is 10-fold ; i.e., 2 points of contact with the great circle, 4 

 intersections with it, and 4 intersections of the two branches. 



This process fails when applied in connection with the second method of 

 § 9, for it brings in triple points which cannot be opened up into three double 

 ones without depriving the whole figure of the desired property. 



12. The 10-fold, whose genesis is described in last section, has the form 

 shown in Plate LXXIX. fig. D, where the great circle is made prominent. It 

 is easily recognised as the ordinary amphicheiral, fig. 31, of Plate LXXX. 

 The reason why it figures in both categories is that the arrangement of the 

 right or left handed meshes, being symmetrical, is not changed by reversion. 

 Thus every ordinary amphicheiral, which is in this sense symmetrical, belongs 

 also to the new kind of amphicheirals with which we are now dealing. 



Plate LXXIX. fig. A shows a 12-fold knot, which is its own inverse with 

 regard to the part drawn as nearly circular, and which is not amphicheiral 

 in the ordinary sense. 



Equal distortions of two corresponding parts give it the new form fig. B, 

 which is also its own inverse with regard to the circular part. 



But if, as in fig. C, we perform one of these distortions alone, the form is no 

 longer its own inverse. But it is certainly amphicheiral, in the sense that it 

 can be distorted into its own perversion. This is effected, of course, by undoing 

 the single distortion which produced C from A, and inflicting the other of the 

 pair of distortions which, together, produced B from A. 



13. Thus there are at least four different senses in which a knot may be 

 amphicheiral. 



A(a) Those in which the systems of right and left hand meshes are similar 

 and congruent. 



A(/3) Unsymmetrical distortions of any of the preceding, when such exist. 

 [When the distortion is symmetrical the knot remains one of A(a).] 



B(a) Those in which the systems of right and left hand meshes are similar 

 but not congruent. 



B(/3) Unsymmetrical distortions of any of the preceding. [When the 

 distortion is symmetrical the knot remains one of B(a).] 



A and B may be spoken of as different Orders, the First and Second ; a and 

 /3 as Classes, First and Second. As already stated, the knot of fig. D belongs 

 to both orders. But no knot can belong either to both classes of one order, 

 or to the first of one order and the second of the other. 



14. In fig. (D) the 10-fold (fig. 31) of § 11 is drawn so as to exhibit its 

 symmetry. And we thus see at a glance that there are at least two ways 



