PROFESSOR TA1T ON KNOTS. 339 



at a point external to each of the fundamental loops, and making each crossing 

 positive when ive first reach it. It is evident that the fundamental loops or 

 coils will now be simply laid on one another. The signs of all the crossings on 

 any one loop may be changed, while that of the base of the loop is immaterial, 

 and this process may be carried out with some or all of the other fundamental 

 loops in any order. Compare the various signs in any state thus produced 

 with those (alternate or not) of the original knot, so as to find the smallest 

 number of changes necessary for its full resolution. The sign of the crossing 

 at the base of each fundamental loop is simply to be disregarded. Another 

 mode of going to work is to alter the signs at pairs of points where two funda- 

 mental loops cross, so as to diminish as far as possible the necessary number of 

 real changes of sign. But we must be very careful in using this process, to see 

 that it does not introduce locking. 



15. When there is locking in part of the knot, the real difficulty is met with 

 only if the crossing or crossings which form as it were the key of the locked 

 part, must also be taken as the base or bases of fundamental loops. In this 

 case we commence the fresh drawing of the knot at a point exterior to the 

 locking, but on the fundamental loop of which one of the key crossings forms 

 the base. This ensures that the completion of the fundamental loop is effected 

 by the last of the operations on the locked part. But the application of the 

 method can be learned far more easily from an example or two than from any 

 rules which could be laid down. Thus the following drawings represent the 

 results of this method as applied to two of the knots already figured. In the 



first of these the two lower external crossings are taken for the fundamental 

 loops, and we see that the knot (if originally over and under alternately) re- 

 quires for its full resolution only the change of sign of each of the two cross- 

 ings which lie in its axis of symmetry. But, if we had chosen the crossings 

 last mentioned as bases of fundamental loops, we should at once have felt the 

 difficulty due to locking. 



In the second, all four crossings in the axis of symmetry close fundamental 

 loops ; but the change of the sign of the lowest of these, alone (which is the 

 key of the locked part), is required for the full resolution. 



