( 235 ) 
XI. — On Knots, with a Census of the Amphicheirals with Twelve Crossings. 
By Mary Gertrude Haseman. Communicated by Dr C. G. Knott, General 
Secretary. (With One Plate.) 
(MS. received March 28, 1917. Read June 4, 1917. Issued separately February 1, 1918.) 
The theory of the knotting of curves, except for a few elementary theorems due 
to Listing,* was entirely neglected until Tait f was led to a consideration of knots 
by Sir W. Thomson’s (Lord Kelvin’s) work on the Theory of Vortex Atoms. He 
attacked chiefly the problem J of constructing knots with any number of crossings, 
and obtained a census of the knots of not more than ten crossings. Those knots 
which exhibit a special kind of symmetry — the amphicheiral knots — offer certain 
points of interest. 
§1. Knot Schemes. 
Tait has introduced two schemes for representing knots : the alphabetical and 
compartment symbols. 
Alphabetical Symbol. — The alphabetical scheme of a knot is based upon the idea 
of the sequence of the crossings which exist on the plane projection of the knot. In 
the case of the alternating knot, the thread passes alternately over and under at the 
crossings. It is convenient to distinguish between the over and under crossings by 
means of the signs + and — respectively. Starting with an over crossing a , the 
alternate crossings may be denoted by b, c, d, etc. In this way there is obtained a 
+ + + 
definite sequence of the letters a, b, c, arranged so that those occupying the odd 
places represent over crossings, while those in the even places denote under crossings. 
Thus Tait’s problem of constructing the plane knots with n crossings reduced itself 
to a question of the essentially different ways in which the even places of the sequence 
a b c .... n 
+ + + + 
may be filled in with the same letters so as to form unipartite closed curves. For 
example, the only arrangement in the case of three crossings is 
acbacb^a 
Hence the “ trefoil ” knot is the only knot of order 3. 
Compartment Symbol. — Tait obtained his idea of the compartment symbol from 
the Listing type-symbol, which depends upon the division of the plane into n + 2 
* Listing, Vorstudien zur Topologie (1874). 
t Tait, Trans. Roy. Soc. Min., xxviii (1876-77), pp. 145-191 ; xxxii (1882-86), pp. 327-342, 493-506. See also 
Scientific Papers, vol. i, pp. 273-347. 
} The same problem has been considered by Kirkman, Trans. Roy. Soc. Edin., xxxii (1882-86), pp. 281-309 ; 
and by Little, Proc. Gonn. Academy, vii (1885-88), pp. 27-43. 
TRANS. ROY. SOC. EDIN., VOL. LII, PART I (NO. 11). 
38 
