( 253 ) 



IX. — Alternate ± Knots of Order Eleven. By Professor C. N. Little. 



(With Two Plates.) 



(Read 21st July ; Revised December 1890.) 



1. A year ago last April, Prof. Tait proposed that I should undertake to derive 

 from Mr Kirkman's polyhedral drawings the alternate ± knots of eleven crossings, 

 thus doing for order 11 what had been clone so admirably by himself in orders 8, 9, 

 and 10. 



2. The work has been a very protracted one, because of the great number of forms 

 involved — more than three times as many as in all preceding orders combined. Mr 

 Kirkman's manuscript contains 1581 forms, of which 22 are bifilar and 16 duplicates. 

 I find from the remainder 357 knots with 1595 forms as shown in the following 

 table : — 



Class. 



£ 



6 



CD 

 Sh 



H 



O 

 fa 



cd" 

 > 





e 



cd 

 > 



CD 



m 



bo 

 fa 



a5 



d 



pi 



CD 

 !> 

 CD 



fa 



CD 

 j> 



a 



CD 

 CD 



S3 

 O 

 fa 



CD 

 CD 

 eg 



fa 



n 



CD 

 CD 



CO 



CD 

 CD 



bo 

 fa 



J3 

 CD 





II 

 III. 

 IV. 



V. 

 VI. 



1 



4 



8 



26 



44 



14 

 25 



48 



3 



6 



18 



6 

 14 

 26 



... 

 l 



4 



... 



2 

 19 

 17 



1 

 1 



8 

 15 



5 



2 

 3 



"l 



3 

 16 



1 



2 



1 



3 



6 



3 





Total > 

 Knots ) 



83 



87 



27 | 46 



5 38 



2 23 



5 



5 



1 



19 



1 ! 2 



4 



6 



3 



357 



As this is an odd order, perversion doubles these numbers, making 714 elevenfold 

 knots, with crossings alternately over and under. 



3. It has been thought unnecessary to show upon the Plates more than one form 

 of each knot ; all, however, have been drawn. Knots of each class having the same 

 number of forms are grouped together to make more simple the identification of a 

 particular elevenfold. A small figure following the series number upon the plates 

 indicates how many distinct forms each knot can assume. Knots 84, 357, and 238 6 are 

 misplaced. 



4. Below each knot-form figured will be found the number of the corresponding 

 form in Mr Kirkman's manuscript, and partition symbols to which the following table 

 gives the key : — 



VOL. XXXVI. PART II. (NO. 9.) . 2 Q 



