CONSTRUCTION OF KNOTS OF FEWER THAN TEN CROSSINGS. 



307 



I think that no difficulty will present itself in the construction of higher 

 w-folcl knots, which has not been met in the preceding pages. 



Here follow the abbreviations used in the descriptions of symmetry : — 



Moncli. for Monarchaxine. 

 Triax. or Tri. for Triaxine. 

 Triarch. for Triarchaxine. 

 Zo. for Zoned. 

 Az. for Azonal, or Zoneless. 



Mox. for Monaxine. 



Mo?,, for Monozone. 



Horn, for Homozone. 



Hct. for Heteroid, not janal. 



2p. for 2-ple, repetition about an axis. 



POSTSCRIPT, September 1, 1884. 



65. As it is a brief matter, it may be worth the while to show how all solid 

 knots can be constructed without omission or repetition. 



Solid Knots, Prime and Non-prime. — A solid knot Q of n crossings is prime 

 or non-prime according as it has or has not a crossing or summit A3B3, A and 

 B being any meshes. 



Lowest Triangle of a Solid Knot Q. — It is easily proved that no solid knot 

 has fewer than eight triangles. The triangle L of Q is ABC , DEF , where 

 ABC are the three covertical faces and DEF the collaterals of L, the lesser 

 being written before the greater in both triplets. 



If L' be another triangle of Q, the lower of LL' has the least A, whatever 

 be the other five faces. If A = A', the lower has the least B. If also B = B', 

 the lower has the least C. If ABC and A'B'C are alike, the lower has the 

 least D, and so on. 



If the six faces are alike in both, it is wisest, and almost sure to be right in 

 construction, to presume that L and L' are identical, or one the reflected image 

 of the other, by the symmetry of Q, which is soon decided. If they are not thus 

 proved alike, an examination of the collaterals of ABCDEF cannot fail to 

 determine the lower triangle. The one whose A, or, if required, whose B, &c, 

 has the lowest collaterals is the lower. 



66. Reduction of a Solid Knot Q. — The simple rule is, efface the edges of 

 the lowest triangle L of Q, or of a lowest when Q has a symmetry. 



Such reduction of a prime solid knot Q of n crossings gives us a subsolid or 

 an unsolid knot P of n-3 crossings, which has one, two, or three flaps, 

 according as the effaced L had one, two, or three covertical triangles : and L 

 must have one, or, by our first definition (65), it cannot be lowest on Q. 



Such reduction of a non-prime Q gives either a non-prime P' or a prime 

 solid knot P of n-2 crossings ; but I am not certain that this can ever be P'. 



