of Edinburgh , Session 1876-77. 
313 
are employed to mark off separate parts of a knot, it is either (a) 
virtually unconstrained, or ( b ) it is divisible (and is actually divided) 
into two separate knots.] If, under these circumstances, changes 
can he made on one side of the fixtures, they will be practicable 
also in whatever way these points be connected by the rest of the string 
provided always that it be not led through the amphicheiral part. 
Hence, if there be an amphicheiral part of any knot, it may often 
be transformed in situ ; the rest of the knot being unaltered, but 
the amphicheiral part being made to present, as it were, another 
hand to the rest. 
To begin with a very simple case, let us take the simplest amphi- 
cheiral form — the complete knot with four crossings—as below. 
1 . 2 . 
When the first of these is inverted, 0 being within either of the 
interior three-sided meshes, the second is produced : if 0 be in one 
of the boundary three-side meshes the perversion of the second is 
produced. But we may easily convert 1 into 2 by fixing the lower 
crossing (i.e., by fixing a point near it in each of the four lines 
diverging from it, so that the dotted part remains fixed), and making 
the upper loop rotate so as to banish the upper crossing* Thus 
the upper parts of these two figures are equivalent. 
And we can now suppose the (dotted) portions between the fixed 
points to be cut open and reknotted in any way we choose, —subject, 
of course, always to the rule of alternate over and under, else the 
knot would in general be reducible. Of course, unless we wish to 
study linking , the ends must be joined so as to preserve con- 
tinuity throughout the string. 
The simplest modes of joining (without additional intersections) 
give us at once two different aspects of the same “ trefoil 5 ’ knot : 
— with one crossing additional we have the original figures 
