UNIFILAR KNOTS OF TEN CROSSINGS. 



485 



tiguous loops of a (2 + «)-ple flap (i>0), the knot can be reduced to an unifilar, 

 solid or unsolid, of n-2 crossings by shrinking up those two loops to a point, 



It is evident that, if any clear definition of a leading flap and of a fixed 

 flap be made and stuck to, the constructing converses (easily defined) of these 

 three theorems must completely solve the following problem : — 



The non-compound unifilars of n-1 and of n-2 crossings, alternately over 

 and under, being given, to construct all the unifilars K„ above defined of 

 n crossings, without risk of repeating a result in any posture, or of making a 

 plurifil knot. 



All that Ave have to do in reducing K„ is to do that at a leading or co-leading 

 flap. All that we have to do in constructing K„ on a base, is to see that we 

 do it by drawing or completing a flap which shall be the leader, or a co-leader 

 on K„. And we shall of course define that a plural flap leads any single one. 



Thus, by theorem A, 4 A (vide PI. XL. vol. xxxii. Trans. R. S. E.) reduces 

 to 3 A, on which it is regularly built by its even flap. 



By theorem B, 6 A reduces to 4 A, on which 6 A is properly constructed by 

 its odd flap. 



By theorem C, C F and 6 G reduce to 4 A, on which by a double flap either is 

 correctly formed. 



By these little examples the constructing converses are plainly suggested. 



This appears to make an end of the puzzle of unifilar knots whose crossings 

 are all through alternately under and over, so far as their construction upon 

 lower non-compound unifilars is desired, as a preparation for the curious 

 transformations and reductions by twisting of Listing and Tait. 



I fear that my distinction of subsolids and unsolids is of little value, as a 

 subsolid can often be twisted into an unsolid, and vice versa. 



I have had theorems B and C for nearly a year. Had I obtained theorem A 

 earlier, my tasks on the unifilars of 8, 9, 10, and 11 crossings would have been 

 much easier, and under less risk of error. 



The simplicity of the three theorems is provoking enough, as usual, after 

 the labour spent with clumsier tools, which looked so much more learned. 







F 1= =l. 





1. 



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