360 
Proceedings of the Royal Society 
an even 2-gon, can be reduced to an unifilar of n - 1 crossings 
by shrinking up that 2-gon to a point. 
4. Let the sequence of 2 n terms 
128 .. . al23 . . . 5 . . . (C) 
be the circle of an unifilar n P of n crossings. 
This n P has an odd 2-gon 12; for since we pass over both its 
threads from 1 to 2, it must have one dot within and one without 
it. Any unifilar that has an odd 2-gon is represented by (C). We 
now write this, omitting the duads 12 and no other, and simply 
reversing 3 . . . 5 — one sequence between 2 and 1 ; 
28 . . . al5 ... 3 . . . (D). 
What means this, when from the projection of M P we have deleted 
the two edges, and consequently the two crossings of the 2-gon 12? 
5. The edges of the crossings 1 and 2 in (C) are 12, la, 12, 15 
and 21, 28, 21, 23. These make angles 212 and 51a vertically 
opposite, and 121 and 823 vertically opposite, where 51a and 823 
are angles of the (4 + r)-gon F' laid bare by deletion of the 2-gon 12. 
The points 1 and 2 of these angles are merely bends or creases at 
the mid-points of the edges 5a and 83 of the (2 + r)-gon F. Effacing 
the creases 1 and 2, (D) becomes 
8 ... a5 ... 3, . . . (E); 
which contains every crossing of (C) but 1 and 2, and every duad 
of (C) except 12, la, 15, 23, 28, and has besides the new edges 83 
and 5 a of the (2 + r)-gon F. This (E) is the circle of an unifilar 
n _ 2 Q of n - 2 crossings. We have thus proved 
Theorem B. — If any unifilar knot W P of n crossings has an odd 
2-gon, the knot is reduced to an unifilar „_ 2 Q of n— 2 crossings by 
the deletion of the two edges, and consequently of the two crossings 
of that 2-gon. 
6. Let 
. . d765r . . . p567s 
. . d765r . . . p7 65s . . . (F) 
be two circles of two unifilars of n crossings. In both of them 56 
and 67 are contiguous 2-gons, having a common crossing 6 — that is, 
both knots have a double 2-gon (a plural flap) 567. Let the first 7 
