ORDERS OF KNOTTINESS 107 



knot. We may suppose it drawn out and flattened until the crossings have 

 been well separated and reduced to the lowest possible number. Projected 

 on the plane this will appear as a closed curve with a certain number of double 

 points. Hence the fundamental mathematical problem may be thus stated : 

 Given the number of its double points, find all the essentially different forms 

 which a closed continuous curve can assume. Beginning at any point of the curve 

 and going round it continuously we pass in succession through all the double 

 points in a certain order. Every point of intersection must be gone through 

 twice, the one crossing (in the case of the knot) being along the branch which 

 passes above, the other along the branch which passes below. If we lay down 

 a haphazard set of points and try to pass through them continuously in the way 

 described, we shall soon find that only certain modes are possible. The 

 problem is to find those modes for any given number of crossings. Let us 

 begin to pass the point A by the over-crossing branch. We shall evidently 

 pass the second point by an under-crossing branch, the third by an over- 

 crossing again, and so on. Calling the first, third, fifth, etc., by the letters 

 A, B, C, etc., we find that after we have exhausted all the intersections the 

 even number crossings will be represented by the same letters interpolated 

 in a certain order. To fulfil the conditions of a real knot, it is clear that 

 neither A nor B can occupy the second place, neither B nor C the fourth, and 

 so on. This at once suggests the purely mathematical problem : How many 

 arrangements are there of n letters when a particular one cannot be in the 

 first or second place, nor another particular one in the third or fourth, nor a 

 third particular one in the fifth or sixth, and so on. Cayley and Thomas Muir 

 both supplied Tait with a purely mathematical solution of this problem ; but 

 even when that is done, there still remain many arrangements which will not 

 form knots, and others which while forming knots are repetitions of forms 

 already obtained. These remarks will give an idea of the difficulties attending 

 the taking of a census of the knots, say, of nine or ten intersections what 

 Tait called knots of nine-fold and ten-fold knottiness. If we take a piece of 

 rubber tubing plaited and then closed in the way suggested above, we shall be 

 surprised at the many apparently different forms a given knot may take by simple 

 deformations. Conversely, what appear to the eye to be different arrangements, 

 become on closer inspection Proteus-like forms of the same. While engaged 

 in this research, Tait came into touch with the Rev. T. P. Kirkman, a 

 mathematician of marked originality, and one of the pioneers in the theory 

 of Groups. Kirkman's intimate knowledge of the properties of polyhedra 



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