‘250 
MARY GERTRUDE HASEMAN : ON KNOTS, 
sufficient importance to be worth separate treatment. Knots of this character have, 
however, so much in common with the regular amphicheirals of the second order, 
that it seems convenient to call them skew amphicheirals of the second order. 
Following Tait’s construction for the special knot in question, consider on a 
sphere a non-singular closed curve that has, in common with a great circle, no 
points except m points of contact, conveniently placed at the alternate vertices 
Vi, V 2 , .... V,, of a regular polygon with 2/* sides; and a similar curve on the 
opposite hemisphere to touch the circle at the remaining vertices V/, V 2 7 , . . . . V/. 
Projection from either pole of the great circle shows that this construction may be 
accomplished in the plane by drawing inside a circle a non-singular closed curve C 
which touches the circle at the alternate vertices v 1} v 2 , .... ^ of a regular polygon, 
of 2m sides and a corresponding curve C 7 , to touch at the remaining vertices 
v/, v 2 , .... v/. When the points of contact are regarded as crossings, the figure 
that results possesses the desired amphicheiral symmetry, although it is not 
necessarily unifilar. The plane projection of such a knot from the mid-point of 
an arc | ViV/ | of the great circle on the tangent plane at the diametrically 
opposite point exhibits symmetry about a point, as in the case of the amphicheirals 
of the first order. 
In the case when m is odd, a point of contact v 1 of the curve C is opposite to 
a point of contact of the curve C 7 , and the curve C is opposite to the curve C 7 . 
Hence the resulting knot is an amphicheiral of the second order. 
On the other hand, if m is even, the point of contact tq is opposite to the point 
u m / 2 . Corresponding arcs are no longer opposite as to the circle. Nevertheless the 
corresponding compartments of the resulting knot are equal and non-congruent, 
as in the amphicheirals of the second order. 
If m= 4, the peculiar eightfold knot given by Tait is obtained. 
The special case m = 0 . mod 3 leads always to a trifilar link. For suppose the 
knot to be described by a point P in a fixed direction, starting from the point v x 
along the arc | uqt> 2 | of the curve C. It leaves this arc at the point v 2 along the 
circle, only to return to the curve C after the elapse of four vertices of the regular 
polygon; that is to say, in going once around the circle, every third arc | v t t] | of the 
curve C is described. If therefore the number of such arcs is a multiple of 3, the 
point P returns to the position rq along the arc | uqi> 2 | by which it left, before 
the complete knot has been described. A second thread of the knot is traversed 
if the point P starts from the point- v 2 along the arc | v 2 v 3 | . And, starting from 
the point v 3 along the arc | u 3 u 4 | , a third thread is obtained, thus completing the 
description of the knot. On the other hand, m = 0 mod Jc, where k is any other 
number, must lead to unifilar knots, since it will be necessary for the point P to 
go around the circle three times before returning to the starting-point along the 
same arc by which it left. The primary compartment symbol for such a knot 
contains 2^+1 compartments, one with m angles and m with three angles each. The 
