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XVIII. — Non- Alternate ± Knots, of Orders Eight and Nine. By C. N. Little, of 

 Nebraska State University. (With a Plate.) 



(Read 15th July 1889.) 



1. To complete the census of knots of any given order, that is, minimum number of 

 crossings, it is necessary to include not only those in which the crossings are taken 

 alternately over and under [Alternate ±), but also the Non-Alternate ±, those in which 

 two or more consecutive crossings are alike over or alike under. Professor Tait has 

 figured the forms of the alternate db knots, of orders three to nine inclusive, on PL XLIV. 

 vol. xxxii., Trans. Roy. Soc. JEdin.; the object of this paper is to describe the non- 

 alternate knots of these orders. 



2. The projection of a non-alternate knot as a single closed line with double points 

 only must be found in the complete series of forms of the alternate knots. The converse 

 is not true, and the first operation is to exclude from consideration all forms which are 

 not projections of non-alternate knots. 



I find it convenient to use* X and y, as shown in the figure, to give the ^ * 

 character of a crossing. The crossing shown looked at as belonging to the d xxxy c 

 compartment, or, briefly, part A or B is a X crossing. Looked at as belonging to £^ 

 C or D, it is a y crossing. 



3. It is to be remembered that in an alternate knot, or in any portion of *a knot 

 where the law of over and under is preserved, the crossings looked at as belonging to the 

 parts of either partition (group of compartments) are alike. 



It is evident that in a coil (succession of 2-gons), all the 2-gon crossings are either 

 lambda or gamma. 



If two parts are opposite at a crossing, and have besides a connection — as a coil — 

 in which all crossings are, say, X, then will the first be a X crossing. It follows that all or 

 none of the forms of an alternate will be included among the forms of a non-alternate knot. 



4. It is now easy to decide whether a given alternate form can be the projection of a 

 non-alternate knot. For example, in the first form of VI., of the nine folds as shown on 

 Professor Tait's plate, the 6-gon amplexum is joined to a 4-gon by a 2-gon, twice by a 

 3-gon and by a single crossing, X say. Each 3-gon connection has its three crossings alike, 

 and X by § 3 above. If now the single crossing be shifted beyond a 3-gon, the 2-gon 

 also is seen to have X crossings, and the law of over and under must hold throughout the 

 form. No form then of this knot VI. can be the projection of a non-alternate knot. 



In like manner it is easily seen that the only forms of PL XLIV. , which are the pro- 

 jections of non-alternate knots, are the following : — 



Eightfold : I, III, IV, VII, IX, and XIV. 



Ninefold: I, III, IV, V, VII, IX, X, XI, XIII, XIX, XXIII, XXIX, XXXV, 

 and XXXVI. 



* Listing called a lambda crossing 8 and used * for a gamma crossing (Vorstudien zur Topologie, p. 52, Gottingen, 1848). 

 VOL. XXXV. PART II. (NO. 18.) 5 P 



