308 
Proceedings of the Royal Society 
than the number of crossings. The sum of the products of the 
coefficients into the corresponding exponents gives, in each of the 
two parts of the symbol, double the number of crossings. These 
symbols contain the topologic character of any particular knotting. 
Listing next points out that the reduced three-crossing knot may 
he obtained by three half-turns about one another of two originally 
parallel cords whose ends are afterwards joined into one ring, 
and that the character depends upon the direction of the torsion. 
He proceeds to give a symmetrical knot, with seven crossings, in 
two different reducible, and one reduced, forms. The reduction of 
the first gives the three crossing knot, that of the second the four 
forms of the essentially non-clear coil of five intersections. (Figs. 
6-9 of my first paper.) 
Their common symbol he writes as 
2 >- 4 + r 2 | 
2/3 + 2/ 2 j , 
and he points out that, in this case, the Amplex belongs to each 
in succession of these four kinds of meshes. 
He then states that the symbol 
2r 4 + 3r 2 ) 
2Z 5 + 2 1 2 I 
gives five different reduced forms, each with seven crossings; while 
the symbol 
r 5 + 3r 3 
Z 4 + 2/3 + 2/ 2 
has six distinct forms. 
But he adds the following extremely important remark : — u In 
certain cases one symbol is equivalent to another, so that the 
reduced forms of the one can be transformed into those of the other.” 
He states that this is the case with the last written symbol and the 
following: — 
2r 4 + 2/’ 3 
/ 4 + 2/3 + 21 “ 
which has five reduced forms. 
Thus there are, in all, eleven reduced forms of these kinds, all 
equivalent to one another, and all having seven crossings. The 
