MARY GERTRUDE HASEMAN ON AMPHICHEIRAL KNOTS. 
599 
possible, however, to construct on the sphere one of the amphicheiral forms of 
the knot given in fig. 2 by means of a curve Ci in contact with a great circle 
at two diametrically opposite points, Pi, P 2 , and of a curve C 2 which is obtained 
as a reflection of C! in the plane tt x of the great circle, followed by a second 
reflection in a plane tt 2 passed through the points Pi, P 2 perpendicularly to the 
plane 7 iq. To secure this construction in the plane suppose c x , which is the 
projection of the curve Ci in the plane tt 1} to be the broken curve in fig. 4, 
with contacts at the diametral points p x , The curve C 2 , represented by the 
dotted curve, is the same curve as c x , but drawn on the outside of the circle and 
reflected in the line p x p 2 - Now, imagine the curve c 2 to be rotated through an angle 
of - to the right or left ; the resulting knot, where the contacts are regarded as 
4 & ’ O’ o _ 
crossings, is found to be the knot shown in fig. 3. 
The foregoing construction led me to seek for a similar construction in the plane 
of the amphicheirals of the first order with any number of crossings. Let the curve 
Fig. 2. Fig. 3. 
Ci have k contacts, 2 < intersections with the circle and <r self sections ; and denote by 
c 2 the same curve on the outside of the circle but reflected in a diameter passing 
through one of the contacts. Now, imagine c 2 rotated through angles — , etc., and 
the resulting curve is found to be an amphicheiral of order 1 with 2(/c + 2* + cr) cross- 
ings. The curve c 2 , obtained by the reflection in the plane tt x of the great circle, 
ensures the desired correspondence of compartments ; the reflection of the curve c 2 
in the plane tt 2 does not alter the number of compartments, nor the number of laps 
of thread bounding each compartment ; instead it interchanges corresponding 
adjacent compartments. For instance, suppose the two regions p x , p 2 by the first 
reflection to go into the two adjacent regions p x , p 2 respectively. If, now, by the 
second reflection p x becomes adjacent to p 2 , then p 2 must become adjacent to p x by 
the same reflection. Since rotation through an angle merely adds one crossing to 
each of the regions which are in the relation of p x , p 2 , then the desired correspond- 
ence of compartments remains unaltered. Further corresponding crossings occur at 
equal arcual distances from the mid-points of the lap of thread common to two 
corresponding compartments. Hence the knot is an amphicheiral of the first order. 
