Prof. Tait on Listing's Topologie. 45 



It is greatly to be desired that some one, with the requisite 

 leisure, should try to extend this list, if possible up to 11, as 

 the next prime number. The labour, great as it would be, 

 would not bear comparison with that of the calculation of ir 

 to 600 places, and it would certainly be much more useful. 



Besides, it is probable that modern methods of analysis 

 may enable us (by a single " happy thought " as it were) to 

 avoid the larger part of the labour. It is in matters like this 

 that we have the true "raison d'etre " of mathematicians. 



(23) There is one very curious point about knots which, 

 so far as I know, has as yet no analogue elsewhere. In 

 general the perversion of a knot (i. e. its image in a plane 

 mirror) is non-congruent with the knot itself. Thus, as in 

 fact Listing points out, it is impossible to change even the 

 simple form (fig. 14) into its image (fig. 15). But I have 

 shown that there is at least one form, for every even number 

 of crossings, which is congruent with its own perversion. 

 The unique form with four crossings gave me the first hint 

 of this curious fact. Take one of the larger laps of fig. 17, 

 and turn it over the rest of the knot, fig. 18 (which is the 

 perversion) will be produced. 



We see its nature better from the following process (one of 

 an infinite number) for forming Amphicheiral knots. Knot 

 a cord as in fig. 19, the number of complete figures of 

 " eight " being at pleasure. Turn the figure upside down, 

 and it is seen to be merely its own image. Hence, when the 

 ends are joined, it forms a knot which is congruent with its 

 own perversion. 



(24) The general treatment of links is, unless the separate 

 cords be also knotted, much simpler than that of knots — i. e. 

 the measurement of belinkedness is far easier than that of 

 beknottedness. 



I believe the explanation of this curious result to lie mainly 

 in the fact that it is possible to interweave three or more con- 

 tinuous cords, so that they cannot be separated, and yet no one 

 shall be knotted, nor any tico linked together. 



This is obvious at once from the simplest possible case, 

 shown in fig. 20. Here the three rings are not linked but 

 locked together. 



Now mere linkings and mere lockings are very easy to 

 study. But the various loops of a knot may be linked or 

 locked with one another. Thus the full study of a knot re- 

 quires in general the consideration of linking and locking 

 also. 



(25) But it is time to close, in spite of the special interest 

 of this part of the subject. And I have left myself barely 



