PROFESSOR TAIT ON KNOTS. 177 
Il. The knot figured in Plate XVI. fig. 4 has no beknottedness. 
III. That in fig. 5 is reducible to the trefoil. 
These are left as exercises to the reader. 
Part IV. 
Beknottedness. 
-§ 35. Recurring to the two species of five-crossing knots discussed in § 18, 
we easily see that there is less entanglement or complication in the first species 
than in the second. For if the sign of ether of the two crossings towards the 
top of the first figure be changed, it is obvious that it will no longer 
possess any but nugatory crossings. But if we change the sign of any one 
crossing in the pentacle, that crossing, and one only of the adjacent ones, become 
nugatory, so that the knot becomes the trefoil with alternating + and —. 
This, in turn, has .all its intersections made nugatory by the change of sign of 
any one of them. Thus one change of sign removes all the knotting from the 
first of these knots, but two changes are required for the second. 
In what follows the term eknottedness will be used to signify the peculiar 
. property in which knots, even when of the same order of knottiness, may thus 
differ : and we may define it, at least provisionally, as the smallest number of 
changes of sign which will render all the crossings in a given scheme nugatory. 
This question is, as we shall soon see, a delicate and difficult one. It is 
probable that it will not be thoroughly treated until one considers along with it 
another property, which may be called Anotfulness—to indicate the number of 
knots of lower orders (whether interlinked or not) of which a given knot is in 
many cases built up. But this term will not be introduced in the present paper. 
§ 36. It may be well to premise a few lemmas which will be found useful 
in examining for our present purpose the plane projection of a knot. 
(a). Regarding the projection as a wall dividing the plane into a number of 
fields, if we walk along the wall and drop a coin into each field as we reach it, 
each field will get as many coins as it has corners, but those fields only will have 
a coin in each corner whose sides are all described in the same direction round. 
For we enter by one end of each side and leave by the other. The number of 
coins is four times the number of intersections ; and two coins are in each cornet 
bounded by sides by each of which we enter, none in those bounded by sides 
by each of which we leave. Hence a mesh, or compartment, which has a coin 
in each corner has all its sides taken in the same direction round ; and we see 
by fig. 6, Plate XVI., that this is the case with twists in which the laps of the 
cord run opposite ways, not if they run the same way. Compare this with the 
remarks of § 35, as to the two species of 5-fold knottiness. 
VOL. XXVIII. PART I. 2 Z 
