( 771 ) 



XXX. — Non-Alternate =b Knots. By Professor C. N. Little, Ph.D. 



by Professor Tait. (With Three Plates.) 



Communicated 



(Read July 3, 1899.) 



1. The following paper is a contribution to the theory of non-alternate ± knots, 

 together with a census of these knots for Order Ten ; that is, all the knots are given 

 which have in reduced form just ten crossings, and in which the thread does not proceed 

 alternately over and under. 



The census was begun in the fall of '93, and carried so far that the forms were 

 drawn. The matter was then laid aside and taken up anew in the spring of '99. 



2. Having postulated an endless one-dimensional continuum which may change its 

 length and form in any way, subject only to the condition that it can never have a 

 double point, and consequently no one portion can be made to break through another, 

 1 understand by a knot a continuum ivliich can not be brought to a circular form. 



3. The above definition makes of a knot a purely mathematical concept. As 

 physical approximations for the continuum may be taken : — (a) a vortex filament of 

 frictionless ether; (6) a flexible, extensible thread. It is convenient and can lead to no 

 error, although keeping to the mathematical conception, nevertheless to speak of the 

 continuum as a thread. 



4. Through the work of Listing,* Tait,! Kirkman,J and the writer, § but pre- 



* "Vorstudien zur Topologie," Gottinger Studien, 1847, pp. 859-866. To the kind courtesy of Professors Felix 

 Klein and P. Stackel, for which I here express my appreciation, I owe the opportunity to examine the topological 

 Nachlasse of Gauss and Listing. The former will appear in the forthcoming Bd. VIII., Gesammelte Werke, and must 

 not be commented upon in advance of publication. The latter contains among the drawings of reduced knots not 

 figured in the "Vorstudien," a sheet bearing date March 18, 1849, on which are the following forms marked as 

 equivalent : — 



j23 3 +3 2 



The interest of this series lies in the fact that it shows that Professor Listing fifty years ago recognised the 

 amphicheiral character of this knot. 



+ These Transactions, vol. xxviii. pp. 145-190 ; vol. xxxii. pp. 327-342 ; ibid., pp. 493-506. See also Collected 

 Scientific Papers, vol. i. pp. 273-346. 



| These Transactions, vol. xxxii. pp. 281-309 ; ibid., pp. 483-491. 



§\Trans. Connecticut Academy, vol. vii. pp. 1-17 ; these Transactions, vol. xxxvi. pp. 253-255. 



VOL. XXXIX. PART III. (NO. 30). 6 E 



