


172 PROFESSOR TAIT ON KNOTS. 
§ 27. A question closely connected with plaited clear coils is that of the 
numbers of possible arrangements of given numbers of intersections in which 
the cyclical order of the letters in the 2d, 4th, 6th, &c., places of the scheme 
shall be the same as that in the Ist, 3d, 5th, &c., @e¢., the alphabetical. 
Instances of such have already been given above. In the first scheme of § 5, 
for example, the letters in the even places are 
DEABC. 
Here the cyclical order of the alphabet is maintained, but A is postponed by 
two places. It is easy to see that the following statements are true. 
Whatever be the number of intersections a postponement of vo places leads 
to nugatory results. 
A postponement of one place is possible for three and for four intersections 
only. 
Postponement of two places is possible only for (four), five, and eight— 
three for seven and ten—four for nine and fourteen—five for (eight), eleven and 
sixteen,—six for (ten), thirteen, and twenty, &c. Generally there are in all 
cases m postponements for 27+1 intersections; and for 32+2, or 3n+1 in- 
tersections, according as n is even or odd. The numbers which are italicised 
and put in brackets above, arise from the fact that a postponement of 7 places, 
when there are 7 intersections, gives the same result as a postponement of 
n—r—1 places. [It will be observed that this cyclical order of the letters in 
the even places is possible for any number of intersections which is not 6 ora 
multiple of 6. | 
When there are 2 postponements with 27+ 1 intersections the curve is the 
symmetrical double coil, z.¢., the plait is a simple twvsz. 
The case with 3n+2 or 32+1 intersections is a clear coil of three turns, 
corresponding to a regular plait of three strands. 
Figures 16, 17 of Plate XV. give the diagrams corresponding to the latter 
case for n=2, 3 respectively; 7.¢., with 8 and 10 crossings. The diagrams 15 
and 18, constructed according to the same plan for 6 and 12 intersections, show 
why there are no multiples of six im this form of coil. In fact, whenever the 
number of crossings in this three-ply plait is a multiple of 6, the strands are 
separate closed curves. 
Part III. 
Methods of Reduction. 
§ 28. Before taking up the question of the complexity of a knot, a word or 
two must be said about the methods of reducing any given knot to its simplest 
form. I have not been able as yet to find any general method of doing this, 
