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XXIII. — Amphicheiral Knots. By Mary Gertrude Haseman, Ph.D. Communicated 
by Dr C. G. Knott, General Secretary. (With One Plate.) 
(MS. received May 20, 1918. Read November 4, 1918. Issued separately December 22, 1919.) 
§ 1. The Intrinsic Symbol of an Amphicheiral. 
The intrinsic symbol* of an amphicheiral knot is based on the idea of the 
sequence of the crossings ; it replaces each letter of the alphabetical symbol by a 
number equal to one-half of the number of crossings intervening before the next 
occurrence of that letter as the knot is traversed in a definite direction. Hence it 
is seen that two knots, which have the same intrinsic symbol, are identical. Since 
the same number of pairs of crossings must elapse before the next occurrence of 
corresponding crossings of two identical knots when the knots are traversed in a 
given direction, it is seen that the converse is true also. It may be necessary to 
consider the complementary intrinsic symbol in order to detect identical knots. For 
example, the two knots 
( 1 ) 10 5599993444488 10 5599993444488 
a g b l c in d h i j f k g b h n e i j f It a l c m d n e , 
( 2 ) 3559 999 10 444488 3559999 10 44448. 8 
a g b l cm d aejfkgbhniejfk li l cm d n i, 
are found to be identical since the intrinsic symbol of (l) coincides with the com- 
plementary symbol of (2). That they are identical may be verified by the fact that 
their compartment symbol is 
By an interchange of the two crossings, a and h in (2), it is seen that the two 
symbols may be made to coincide. So two amphicheiral knots are identical when their 
intrinsic symbols agree except for an interchange of complementary numbers. 
The intrinsic symbol of an amphicheiral knot of the first order offers certain 
points of interest. An amphicheiral centre of an amphicheiral knot of order 1 is 
* M. G. Haseman, Trans. Roy. Soc. Edin., vol. lii, p. 235. 
TRANS. ROY. SOC. EDIN., YOL. LII, PART III (NO. 23). 
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