NON-ALTERNATE zfc KNOTS. 775 



the twist of the perversion of a form is the negative of the twist of the original ; hence 

 the twist of an amphicheiral must be zero, since it must at the same time change sign 

 and be constant. 



Census of Tenfold Non- Alternates. 



11. The basis of a non-alternate census of any order is the corresponding census 

 of alternates of the same order, since projections of the former, looked at as plane 

 curves with double points, must be included in the projections of the latter, also 

 so regarded. 



12. It will be remembered that two censuses* of tenfold alternates were made 

 independently. These agreed in classes TIL and V. The one discrepancy in class IV. 

 was due to an oversight of Professor Tait, and by him corrected before his plates were 

 printed. There were four discrepancies in class VI., for all of which I was responsible. 

 The substantial agreement of these results, obtained by quite different methods, leaves 

 no doubt that we have all the tenfold alternates without error or omission. 



Because of their greater elegance, I have used Professor Tait's drawings upon which 

 to mark the crossings ; the numbers given in the following sections and on the plates 

 are those of the knots of his census 



13. The alternate tenfold knots which can in projection also be the projection of 

 non-alternate knots are the following : — 



(a) Those in which two parts have three necessarily alternate connections : — 



2 39 57 75 115 



5 42 63 100 116 



18 47 64 103 118 



23 50 69 109 123 



36 56 70 114 



The two or three forms of each of these were at once drawn and marked. For 

 example, 123 has two 3-gons connected by the alternate connections 2-gon, 2 2 coil and 

 2 4 coil. Any one of these connections may be y (or A) and the other two X (or 7), 

 giving six forms — the three found on the plates in knots : I, class I. ; I, class III. ; 

 and I, class VI., and their perversions. 



(b) More complicated : — 



6 



27 



38 



86 



95 



8 



28 



44 



87 



96 



9 



29 



46 



88 



97 



10 



30 



54 



89 



98 



12 



31 



59 



90 



99 



13 



32 



82 



91 



110 



17 



33 



83 



92 



111 



25 



34 



84 



93 



112 



26 



35 



85 



94 



113 



* Trans. R.S.E., vol. xxxii. ; Trans. Conn. Acad., vol. vii. 



