240 
MARY GERTRUDE HASEMAN : ON KNOTS, 
Consider the knot given by the alphabetical symbol : — 
(1) afbg.cjdhebf a giheicjd^a 
The symbol shows the possibility of the following distortions : — 
Do° r/ ; . ... a f .... f a ... . 
vT* ..... f b .... b / ... . 
Do”'; . ... cj .... c j ... . 
Do 01 '" ; . ... j d ... . j d ... . 
h - + + - 
D 0 ; .... h e .... he ... . 
D*, : .... h e .... i li e i .. . 
Dj m d ; . ... cj d ... c j d ... . 
D 2 ; .... a / b g .. b f a g .. . 
Of these distortions only Di and D 2 can produce a change of form, since a knot is 
invariant under a distortion D 0 or a continuation of distortions D 0 ; that is to say, 
The application of the distortions Dl, D 2 produce the symbols : — 
j d % h e i' b / a g h e c j d I a 
D x 
( 2 ) 
and 
( 3 ) 
a / b g 
+ ~ + - 
g' a f b c j d h e g' b f a i h e i c j d I a 
respectively. The distortion DxD'!, that is to say the distortion D x , followed by the 
distortion D 2 ’ gives the symbol 
(4) g' a f b c j d i' h e 1 g' b f a h e c j d | g 
While it is a simple matter to recognise in any alphabetical scheme the existence 
of reversible tangles, and the effect thereon of the corresponding distortions, D„, 
it is not so easy to say whether the distorted form and the original are the same or 
different. To meet this difficulty, the alphabetical symbol may be replaced by an 
equivalent numerical symbol * in which for each letter is substituted a number 
equal to one-half the number of crossings intervening before the next occurrence of 
the letter as the knot is described. For some purposes it is convenient to write also 
in a second row the numbers that arise when the knot is described in the reverse 
direction ; but for a knot of order n, the sum of the numbers immediately above and 
below any letter is equal to n— 1. Thus the symbol (l) becomes : — 
(1 # ) 54346663365451668333 
45653336634548331666 
or more simply 
5434666336545 1 6683 3 3 
* Suggested by Professor C. A. Scott, who calls it the intrinsic symbol. 
