PROFESSOR TAIT ON KNOTS. 165 
trefoil knot. It will be seen that in none of these is the connection merely 
apparent, the trefoil part having its signs alternately + and — if those of the 
complete knot have this alternation. But if, for instance, we had drawn the 
fine line horizontally through the trefoil, so as to divide each of the upper two- 
cornered compartments into two three-cornered ones, we should have got No. 
II. of the 7-fold forms, and the original trefoil would have been rendered only 
apparent. 
§ 20. In my British Association paper, already: referred to, I showed that any 
closed plane curve, or set of closed plane curves, provided there be nothing 
higher than double points, divides the plane into spaces which may be coloured 
black and white alternately, like the squares of a chess-board, or, to take a 
closer analogy, as the adjacent elevated and depressed regions of a vibrating 
plate, separated from one another by the nodal lines (Plate XV. figs. 9 and 10). 
I afterwards found that Listinc had employed in his notation for knots, in 
which the crossings are alternately over and under, a representation which 
comes practically to the same thing; depending as it does on the fact that in 
such a knot all the angles in each compartment are either right or left-handed, 
and that these right and left-handed compartments alternate as do my black 
and white ones. 
I have since employed a method, based on the above proposition, as a mode 
of symbolising the form of the projections of a knot, altogether independent of 
its reducibility. I was led to this by finding that Listine’s notation, though 
expressly confined to reduced knots, in which each compartment has all its 
angles of the same character, is ambiguous: in the sense that a Type-Symbol, 
as he calls it, may in certain cases not only stand for a linkage as well as a knot, 
but may even stand for two quite different reduced knots incapable of being 
transformed into one another.* The scheme, already described, has no such 
ambiguity, but it is much less easy to use in the classification of knots. Hence, 
following Listine, I give the number of corners of each compartment, but, 
unlike him, only of those which are black or of those which are white. But I 
connect these in the diagram by lines which show how they fit into one another 
in the figure of the knot. An inspection of Plate XV. figs. 11 and 12 (species 
VII. of sevenfold knottiness) will show at once how diagrams are arrived at, 
either of which fully expresses the projection of the knot in question by 
means of the black or of the white spaces singly. The connecting lines in the 
diagrams evidently stand for the crossings in the projection, and thus, of course, 
either diagram can be formed by mere inspection of the other,t and the rule for 
* Proc. R. S. E. 1877, p. 310 (footnote), and p. 325. 
t Some further illustrations of this will be found in the abstract of my paper on “ Links,” Proc. 
R. S. E. 1877, p. 321. 
VOL. XXVIII. PART I. 7 1} 
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