308 REV. T. P. KIRKMAN ON THE ENUMERATION, DESCRIPTION, AND 



67. Construction of Solid Knots Q of n Crossings. — The rule is the converse 

 of the preceding. Add to the subject P of w-3 crossings, whether P be solid, 

 subsolid, or unsolid, a lowest triangle of the result Q, occupying three mid- 

 edges of a mesh of P. I am* not certain that when P is non-prime such addi- 

 tion can ever be made. 



In order that Q may be solid, P must have fewer than four flaps, which, if 

 more than one, must be collaterals of one mesh. If P has only one flap, it is 

 collateral with two meshes, alike or different. Such a P must not be unsolid. 



It may be that several different lowest triangles of Q may be drawn upon P, 

 giving as many different Q, or that no lowest of Q can be drawn on P. In this 

 case P is no base, and in construction is useless. No knot Q is reducible to it 

 by deletion of a lowest triangle of Q. Examples are given below. 



68. We proceed to construct on our figured knots P every possible solid 

 knot Q. 



Our only knots which have fewer than four flaps, all of which stand about 

 one mesh, are 3 A ; 5 A ; 6 F, 6 J ; 7 A, 7 C ; 8 R, 8 T, 8 W, s Ad, 8 Ag, 8 Ap, 8 Aq, 8 At ; 

 9 A, 9 B, 9 C, 9 I, 9 J, 9 K, 9 L, 9 N, gBm, 9 Bn, 9 D<?, 9 D/, 9 E?/, 9 F/, 9 Gd, Q It ; thirty of 

 them, of which 8 Aq, 9 E?/, 9 F/are not figured, but can easily be drawn (arts. 46, 

 62) on 7 A, 8 T, and 8 W— 



3 A gives the solid 6 J ; 



5 A „ 8 Ap ; 



G F „ Q lt ; see the three figures ; 



7 A „ 10 A , zo. tri. 4 4 3 G , (446) ; 



7 C „ 10 B , 5 zo. monch. horn. 5 2 3 10 , (20) ; 



and also 10 C , az. tri. 4 4 3 8 , (6 , 14) ; 

 8 R gives the solid n A , 2 zo. mox. het. 4 5 3 8 , (6 , 6 , 10) ; 

 s Ad „ U B , 2 zo. mox. het. 5 2 43 10 ,(4,4,14); 



8 At „ n C , moz. 54 3 3 9 , (22) ; 



n A „ 12 A , moz. 54 4 3 9 , (4 , 20) ; 



9 B „ 12 B , az. tri. m s , (24) ; 



9 J „ 12 C,asym. 5 2 4 2 3 w , (24) ; 



12 D, 6 zo. horn. 6 2 3 12 , (888) ; 

 B K „ 12 E, moz. 54*3°, (10, 14) ; 



9 L „ 12 F, 2p. mox. het. 4 6 3 8 , (G, 18) ; 



n N „ 12 G, 3 zo. monch. 4 6 3 8 , (6666) ; 



Bm „ 12 H , zo. triarch. 4 6 3 8 ,(6,6,6,6); 



9 De „ 12 I , 2p. mox. het. 5 2 4 2 3*° , (4 , 4 , 16) ; 



9 Et/ „ 12 J , 2 zo. mox. het. 5 2 4 2 3 10 , (10 , 14) ; 



9 F/ „ 12 K , 2p. mox. moz. 4 6 3 8 , (24) . 



We have thus twenty prime solid knots, of fewer than thirteen crossings, 



