664 C. N. LITTLE ON NON-ALTERNATE =fc KNOTS. 



5. It now becomes necessary to obtain the complete series of forms of non-alternate 

 knots of these orders by assigning to the crossings all possible combinations of X and y. 

 In 8III, 8IX, 8XIV, 9IV, 9XIII, 9XIX, 9XXIII, 9XXIV, and 9XXXVI, two parts are 

 joined by three connections, in each of which all crossings are alike, and the one, two, or 

 three forms can be drawn at once. In each form of 8 III, for example, two parts are 

 connected by a 2-gon and twice by a 3-gon. The two forms have y 2-gon, X 3-gon, 

 y 3-gon ; and X 2-gon, y 3-gon, y 3-gon. Perversion doubles all numbers where 

 amphicheiralism is not found, and will not be again referred to until the final summing up. 

 6. Inspection shows that three consecutive overs cannot exist in these 

 J orders without degradation of the form ; and this consideration greatly 

 shortens the labour of treating the remaining forms. For illustration, 81 

 may be taken. Lettering the form as shown, a table is made out where 

 + and — are used for over and under crossings respectively. A portion of the knot in 

 which the crossings are alike is enclosed in a parenthesis. 



A 



B 



C 



(D 



E) (F G) A 



H 



C 



(F 



G) B 



H 



(D E) 









1 + 



+ 



— 



— 



+ + - - 



+ 



+ 



— 



+ - 



— 



+ - A 



+ 



B + F + 





2 + 



+ 



— 



— 



+ - + - 



— 



+ 



+ 



— — 



+ 



+ - „ 





F - 





3 + 



— 



+ 



+ 



- + - - 



+ 



— 



— 



+ + 



— 



- + A 



+ 



B - C + D 



+ F + 



4 + 



- 



+ 



+ 



- - + - 



+ 



— 



+ 



- + 



— 



- + „ 



5? 



>) J) » 



F - 



5 + 



- 



+ 



— 



+ H 



+ 



— 



— 



+ + 



— 



+ - „ 





D 



- F + 



6 + 



- 



+ 



- 



+ - + - 



+ 



— 



+ 



- + 



- 



+ - „ 





n D 



- F - 



7 + 



- 



- 



+ 



- + - - 



+ 



+ 



— 



+ + 



— 



- + „ 





„ C - F + 





8 + 



- 



— 



+ 



- - + - 



— 



+ 



+ 



- + 



+ 



- + „ 





F - 





From this table the crossings are marked on the forms. No. 6 is the alternate form. 

 No. 3 is an amphicheiral form, but it degrades. No. 7 is the perversion of No. 2, and 

 No. 5 of No. 4. This leaves the four distinct forms: 1, 2, 4, and 8. I find 19 eightfold 

 forms, and for the ninefolds 88 distinct forms. 



7. Thus far, the work has been straightforward and a matter of routine. In deriving 

 the knots from the knot-forms the conditions to be observed are so many that a single 

 worker cannot be absolutely certain that all have been observed, and that all the groups 

 of forms obtained are really distinct knots. 



I find 3 eightfold non-alternate knots, none of which has an amphicheiral form ; so 

 that to the 31 alternate knots already known must be added six new non-alternate, 

 making 37 distinct eightfold knots. Professor Tait has shown at N on PL LXXIX. 

 vol. xxxii., Trans. Roy. Soc. Edin., five forms of knot I. 



In the ninefolds I find 8 non-alternates, and their 8 perversions, making with the 

 82 alternates, 98 ninefold knots. 



These new knots are shown on the Plate. 



8. All of the new knots, with the exception of VIII. of the ninefolds, have single 

 degree of beknottedness ; for in every case it is possible to remove in some form of the 

 knot at least two crossings by changing the sign of one, and without making the form 

 alternate throughout. But there is no non-alternate knot of fewer than eight crossings. 

 Knot VIII. has twofold beknottedness. 



H JUL. 90 



(v A 



