PROFESSOR TAIT ON KNOTS. 337 



projection of PL XLIV. fig. xxxv.) the trefoils are so linked after this 

 operation, that the change of sign of one crossing of either resolves the whole. 



This is, however, much more easily seen by at once changing the signs of the 

 middle and of the lower (or the upper) crossing, for the whole is thus resolved. 

 [This course is at once pointed out by the process of § 13 below, if we choose 

 as fundamental crossings the three highest in the figure.] Hence the beknotted- 

 ness is 1, 2, 2 in the last three figures respectively. 



11. Another instructive example is afforded by the 8-fold knot below, 

 which is figured as iv. on Plate XLIV. : — ■ 



At a first glance it appears to be made of two once-linked trefoils, and there- 

 fore to have three degrees of beknottedness. But a little consideration shows 

 that neither the trefoils nor the link have alternations of signs (i.e., there is 

 neither knotting nor linking), but that the whole is kept from resolution solely 

 by the lap of cord which has been drawn as a straight line in the figure. This 

 forms, as it were, the tail of a Rupert's drop ; break it, and the whole falls to 

 pieces. A change of sign of either of the interior crossings on that lap makes 

 one trefoil ; of either of the 4 lateral external crossings, the 6-fold amphi- 

 cheiral ; of the upper crossing, the 4-fold amphicheiral ; and of the lower axial 

 crossing, the 5-fold of one degree of beknottedness. All these modes of resolu- 

 tion lead to the result that the knot is of 2-fold beknottedness. 



12. It is now obvious why, in consequence of locking and not of amphi- 

 cheiralism as I first thought, the electro-magnetic test fails in certain classes of 

 cases to indicate properly the amount of beknottedness. For it is clear that 

 in pure locking there is no electro-magnetic work along the locked part of any 

 one of the three courses involved. Hence, for the part of a knot or link which 

 is locked, the electro-magnetic test necessarily gives an incorrect indication of 

 beknottedness. Perhaps it may be said that, in such cases, beknottedness is 

 not the proper name for this numerical feature of a knot: — but it is obviously 

 correct if defined as in § 7 above. 



