338 PROFESSOR TAIT ON KNOTS. 



13. A simple but thoroughly practical improvement on the methods given 

 in my first paper for the graphical solution of Gauss' problem (extended) is as 

 follows : — Draw the knot or link, as below, with a double line, like the edges 

 of an untwisted tape, and dot (or go over with a coloured crayon) one of the 



two lines. Now it is easy to see that, of the four angles at a crossing, one 

 angle is bounded by full lines, and its vertical angle by dotted lines. These 

 will be called the symmetrical angles. Also it is clear that the electro-magnetic 

 work has one sign for the crossings when the symmetrical angles are right- 

 handed, and the opposite sign when they are left-handed. Thus we can at once 

 mark each crossing as r or /, silver or copper, at pleasure. If the figure be a 

 knot, and if we cut it along a line dividing a symmetrical angle, re-uniting the 

 pairs of ends on either side of that line, the whole remains a knot (still with 

 alternations of over and under if the original was so), but of knottiness at least 

 one degree lower. When the line divides an unsymmetrical angle, the whole 

 becomes (after re-uniting the ends, as before) two separate closed curves, in 

 general linked and, it may be, individually knotted. [When we treat a link in 

 this way at any of the linkings (i.e., where two different strings cross one 

 another), it becomes a knot. It is curious that by this process a knot is 

 equally likely to be changed into a knot or into a link, while a link always 

 becomes a knot.] This method has the farther advantage of showing at 

 a glance the various sets of crossings which we may choose for omission 

 (in the electro-magnetic reckoning), as due merely to the coiling of the figure, 

 not to knotting, linking, or locking. For each such crossing must belong to a 

 simple loop, which, for reference, we will call fundamental. Such a loop is 

 detected immediately by its having (throughout) the full line or the dotted line 

 for its external boundary, and therefore is necessarily closed at a symmetrical 

 angle. If we now erase these fundamental loops in succession, till no crossings 

 are left, the crossings at their bases form one of the groups which may be tried. 

 When part of the knot has locking, it is sometimes necessary to try more than 

 one of these groups before we arrive at the true measure of beknottedness. 

 As this is a matter of importance, it may be well to discuss it a little farther. 



14. When there is no beknottedness (whether true, or depending on linking 

 or locking), the electro-magnetic work, with the proper correction for mere 

 coiling, is certainly nil. But this proper correction requires to be found, and 

 where there is locking its discovery sometimes presents a little difficulty. 

 When there is no locking, all we need do is to draw the knot afresh, beginning 



