238 
MARY GERTRUDE HASEMAN : ON KNOTS, 
But a distortion # of a non-composite knot is possible if there emerge from a 
limited portion of it, four free ends, of which an adjacent pair is crossed. The four 
free ends indicate that the portion considered is made by a number of crossings of 
two distinct parts of the complete thread, and hence may be called a “ reversible two- 
thread 1 tangle” — or, more simply, a “reversible tangle.” Denote by x, y,p, q the 
four free ends, and let the adjacent pair x, y be crossed at a point a. By a rotation 
through two right angles as above, the two threads x, y are untwisted, and the 
threads p, q are crossed at a point a'. Hence, as Little points out, there will be no 
change in the number of compartments involved in the primary and secondary 
symbols, although the order in which they are joined may be disturbed. 
A distortion is of order n if it involves n + 2 crossings. .Denote by D„ a dis- 
tortion of order n which operates on the crossing a. The reversible tangle R n is 
the portion of the knot which admits the possibility of a distortion D„. 
Fig. 2. Fig. 2'. 
Let the knot be described in such a direction that the reversible tangle R n is 
entered by the thread x over the crossing a, and denote by y 1} y 2 , ... . 7 «+i the 
remaining n + 1 crossings in the order in which they are met. To leave the reversible 
tangle it is necessary either to return to the crossing a and pass out along the thread y, 
or to leave by one of the two remaining threads, say p. Accordingly, the alphabetical 
schemes for the reversible tangle R„ are respectively 
or 
and 
X 
a 7i 
72 • • 
■ ■ 7»+i 
a y . 
■ • • q 7s ■ • 
■ • 7i P 
+ 
+ 
X 
a yi 
72 • • 
• • y n 
ay.. 
■ . ■ P 7, • • 
+ - 
+ 
X 
a 7l 
72 • • 
• • Yn+ 1 
p . . . 
• V a 7i . . 
■7 i 1 
~ 
+ - 
+ 
X 
a 7l 
72 • • 
+ 
• ’ Yn+1 
p • • • 
■17} ■ • ■ 
■ ■ ,7i a 
where y u y^i =f j) denotes the 7 ’s in some order. If a crossing = 1 , 2 , n + l) 
occurs twice on one thread of the reversible tangle R», then only the remaining n 
crossings exist on the second thread. The effect of a distortion on the alphabetical 
* Tait, Trans. Roy Soc. Edin., xxxii (1882-83), p. 328, or Scientific Papers, i, p. 320, recognises the possibility of 
such distortions ; to Little, Proc. Conn. Acad., vii (1885), p. 44, § 10, is due the formulation of necessary conditions 
in the appearance of the knot. 
t In the case of the alternating knot.the possibility of a distortion is limited to the two-thread tangle ; for non- 
alternating knots there may exist distortions of reversible tangles of more than two threads. 
