:530 PROFESSOR TAIT ON KNOTS. 



of this combination depends solely upon the sign of the central crossing. 

 There is no real linking of the two cords, and there is obviously no knotting. 

 But if the sign of any one of the crossings, except the central one, be changed, 

 the whole becomes the simple amphicheiral link, the linking having been 

 Introduced by the change of sign. [This, as will be seen in § 14 below, is an 

 excellent example of a case in which the key-crossing of a locking is also a 

 root-crossing of a fundamental loop.] 



9. We may therefore define, as one degree of locking, any arrangement, or 

 independent part of an arrangement, analogous to that above (whether it be 

 made of one, two, or three separate strings), the criterion being that the change 

 of one sign unlocks the whole. But it is well to notice, again, that if, in the 

 above figure, we change the sign of any crossing except the central one, we 

 have one degree of linking left, and that this has in reality been introduced by 

 the change of sign. This remark extends, with few exceptions, to more 

 complex cases. 



- 10. Thus, though the following 8-fold knot (which I reproduce from 

 Trans. R. S. E., 1877, p. 188) does not, at first sight, appear to depend on 



locking, we have only to make a simple transformation (as ante, § 3) to re- 

 duce it to the symmetrical form in which the single degree of locking is 



at once evident. It was by considering this knot, with its (quite unex- 

 pected) single degree of beknottedness, that I first saw the true bearing 

 of locking in the present subject. (It is given as x. of the 8-folds in Plate 

 XLIV.) 



Other excellent instances of the same difficulty are the following. The first 

 of these is completely resolved, the second changed to the 3-fold, while the third 

 becomes apparently two linked trefoils, all by the change of the single crossing 

 in the middle of the lock. But with the 9-fold knot (which is merely a different 



