PROFESSOR TAIT ON KNOTS. 187 
these over the other must be aneven number (zero included). In general, 
half this number is the beknottedness. But when the knot, or even part of it, 
is amphicheiral there is usually more beknottedness than this rule would give. 
And, in particular, there may be beknottedness when the number is zero. In 
this case the number of silver (and of copper) crossings is even, and is double 
the degree of beknottedness. 
As I have already stated, I have not fully investigated this point, and there- 
fore for the present I content myself with giving two instructive examples from 
the six-fold knots. The observations which will be made on these contain at 
least the germ of the complete solution. 
The form y (of § 8) is not amphicheiral. As there drawn, it has four copper 
and two silver crossings, the latter being the intersections of the loop with the 
trefoil; but the scheme shows that two copper crossings must be omitted from 
the reckoning, one of these being necessarily that which is uppermost in the 
figure. If the sign of this last be changed, the knot opens out, so that it has 
but one degree of beknottedness. Hence, in this case, the two copper and two 
silver crossings correspond to one degree of beknottedness only. But if we 
change the sign of any one of the other three copper junctions the knot sinks 
to'the 4-fold amphicheiral, retaining its one degree of beknottedness. 
In the amphicheiral form £ (of § 8) there are three silver and three copper 
crossings. As the figure is drawn, these are to the right and left of the figure 
respectively ; and either crossing at the end of the lower coil may be left out, along 
with any one of the three on the other side. Thus there remain, as in the former 
case, two silver and two copper ones. This corresponds to one degree of beknot- 
tedness, as in the last case, for the change of sign of ezther crossing at the end of 
the lower coil unlooses the knot. But if any one of the other four crossings 
(alone) have its sign changed, the whole becomes a right or left-handed trefoil 
knot, retaining, as in the former example, its one degree of beknottedness. 
To give the greatest beknottedness to these forms, we must alter two signs 
in (y) and three in (8). In each case one crossing is lost, and the form becomes 
the pentacle (§ 7) with its two degrees of beknottedness. 
Part V. 
Amphichetral Forms. 
§ 47. These have been defined in $17, and several examples have been 
given, not only of knots, but of links, which possess the peculiar property of 
being transformable into their own perversions. 
The partition method (§ 21) suggests the following mode of getting amphi- 
cheirals :—Since the right-handed and left-handed compartments must agree 
