PROFESSOR TAIT ON KNOTS. 



331 



former paper, it may be well to give here the partitions of 2n which correspond 

 to true knots— for the values of n from 3 to 9 inclusive. The various parti- 

 tions, subject to the proper conditions, are all given, in the order of the number 

 of separate parts in each ; those which have a share in one or more of the true 

 knots, as given in the Plate, are printed in larger type. 



n = 3 



n = 6 (contd.) 

 42222 



71 = 8 (contd.) 



772 



n = 9 



n = $ (contd.) 



33 



99 



66222 



222 



33222 



763 



972 



65322 





222222 



754 



963 



64422 







664 



954 



64332 



n = 4 



w = 7 



655 



882 



63333 







8422 



873 



55422 







44 



77 



8332 



864 



55332 



422 



752 



7522 



855 



54432 



332 



743 



7432 



774 



54333 



2222 



662 



7333 



765 



44442 





653 



6622 



666 



44433 





644 



6532 



9522 



822222 



11 = 5 



554 



6442 



9432 



732222 





7322 



6433 



9333 



642222 





55 



6422 



5542 



8622 



633222 



532 



6332 



5533 



8532 



552222 



442 



5522 



5443 



8442 



543222 



433 



5432 



4444 



8433 



533322 



4222 



5333 



82222 



7722 



444222 



3322 



4442 



73222 



7632 



443322 



22222 



4433 



64222 



7542 



433332 





62222 



63322 



7533 



333333 





53222 



55222 



7443 



6222222 



n = 6 



44222 



54322 



6642 



5322222 





43322 



53332 



6633 



4422222 





66 



33332 



44422 



6552 



4332222 



642 



422222 



44332 



6543 



3333222 



633 



332222 



43333 



6444 



42222222 



552 



2222222 



622222 



5553 



33222222 



543 





532222 



5544 



222222222 



444 



11 = 8 



442222 



93222 





6222 





433222 



84222 











5322 



88 



333322 



83322 





4422 



862 



4222222 



75222 





4332 



853 



3322222 



74322 





3333 



844 



22222222 



73332 





The whole numbers of available partitions are thus in order : — 



2, 4, 7, 14, 23, 40, 66. 

 Of these there are employed for knots proper only 



2, 1, 4, 4, 12, 17, 36, 



respectively. The remainder give links, or composite knots, or combinations 

 of these. (See Appendix.) 



To enable the reader to identify, at a glance, any knot of less than 10-fold 

 knottiness, I subjoin the partitions corresponding to each figure in Plate XLIV. 

 It is to be remembered that (as in § 15 of my former paper) deformations which 

 are compatible with the same scheme, however they may change the appearance 



