





184 PROFESSOR TAIT ON KNOTS. 
The 9-fold knot of Plate X VI. fig. 15 has its crossings so drawn as to be 
all copper. Three must be left out of reckoning for the coiling, so it has three 
degrees of beknottedness. 
But if we made the crossings alternately + and — we should find zero for 
the corrected electro-magnetic work—three copper and three silver crossings 
remaining. Change, then, the sign of any one of the three outer or inner 
crossings, and the whole reduces to the 4-fold knot. Hence it has two degrees 
of beknottedness. . 
If the crossing whose sign is changed be neither an outer nor an inner one, 
the result is a very singular 8-fold knot (irreducible, though having continua- 
tions of sign), differing from that of fig. 24, Plate X VL., in the fact that its com- 
ponent trefoil knots are wnsymmetrically linked together. And it has but one 
degree of beknottedness, while that of fig 24 has three. 
I have called attention to this example because of its bearings on the ques- 
tion of the numbers of different irreducible knots having the same projection, 
which we meet with as soon as we reach 8-fold knottiness. 
2. To remove all beknottedness from a projection it is only necessary to 
make every crossing in its scheme + (or —) when it is first met with, reading 
from any point whatever. For then the several laps of the coil are, as it were, 
paid out in succession one over the other. When the beknottedness of a 
scheme so marked is calculated (as in § 41), it will be found that there is 
always at least one choice of a set of crossings such that, when these are 
omitted from the count, the electro-magnetic work is zero. 
As an illustration take the very simplest form, the trefoil knot, with the 
suffixed signs determined by this rule. The scheme is 
+ p> | 
+O+ 
aresh | 
| > | 
| o+ 
| *e0 # 
Ais, 
= 
Now, by § 41 we are entitled to leave out of count either A, B, or C. Leaving 
out either A or B gives zero for the electro-magnetic work, as it ought to be; 
but leaving out C gives — 8 z. 
3. The only way in which we can have the intersections + and — alter 
nately while every letter is + on its first appearance, 7.¢., when there is no 
beknottedness, obviously the wholly nugatory scheme 
A ABB, &c. 
abe 
§ 43. To illustrate these methods let us take again the 5-fold knots in 
§ 18) whose schemes are 
