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XI.—On Knots, with a Census of the Amphicheirals with Twelve Crossings

Published online by Cambridge University Press:  06 July 2012

Mary Gertrude Haseman
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Extract

The theory of the knotting of curves, except for a few elementary theorems due to Listing, was entirely neglected until Tait 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 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.

Type
Research Article
Information
Earth and Environmental Science Transactions of The Royal Society of Edinburgh , Volume 52 , Issue 1 , 1918 , pp. 235 - 255
Copyright
Copyright © Royal Society of Edinburgh 1918

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References

page 235 note * Listing, Vorstudien zur Topologie (1874).

page 235 note † Tait, , Trans. Roy. Soc. Edin., xxviii (1876–1877), pp. 145–191;Google Scholar xxxii (1882–86), pp. 327–342, 493–506. See also Scientific Papers, vol. i, pp. 273–347.

page 235 note ‡ The same problem has been considered by Kirkman, , Trans. Roy. Soc. Edin., xxxii (1882–1886), pp. 281–309;Google Scholar and by Little, , Proc. Conn. Academy, vii (1885–1888), pp. 27–43Google Scholar.

page 238 note * Tait, , Trans. Roy Soc. Edin., xxxii (1882–1883), p. 328,Google Scholar or Scientific Papers, i, p. 320, recognises the possibility of such distortions; to Little, , Proc. Conn. Acad., vii (1885), p. 44,Google Scholar § 10, is due the formulation of necessary conditions in the appearance of the knot.

page 238 note † In the case of the alternating knot the possibility of a distortion is limited to the two-thread tangle; for nonalternating knots there may exist distortions of reversible tangles of more than two threads.

page 241 note * Suggested by Professor C. A. Scott, who calls it the intrinsic symbol.

page 244 note * (Trans. R.S.E., xxxii, p. 494; or Scientific Papers, i, p. 336.) In his third paper Tait deliberately limits himself to this view; but he remarks—” We shall afterwards find that there are at least three other senses in which a knot may be called amphicheiral, and shall thus be led to speak of different orders and classes of amphicheirals.” (See below, § 6.)

page 245 note * Trans. Roy. Soc. Edin., xxxii, pp. 494–497; or Scientific Papers, i, 336–340.

page 246 note * Tait, , Trans. Roy. Soc. Edin., xxxii, p. 496;Google Scholar or Scientific Papers, i, p. 338.

page 248 note * Möbius, , Über die Grundformen der Linien der dritten Ordnung, ii, p. 90Google Scholar.

page 248 note † Cayley, vol. v, op. 361, p. 468.

page 249 note * Trans. Roy. Soc. Edin., xxxii, p. 500; Scientific Papers, i, p. 342.

page 252 note * Tait, , Trans. Roy. Soc. Edin., xxxii, p. 498;Google Scholar or Scientific Papers, i, p. 340.

page 252 note † Tait, , Trans. Roy. Soc. Edin., xxxii, p. 499;Google Scholar or Scientific Papers, i, p. 341.

page 252 note ‡ It must be remembered that if a knot is amphicheiral of the first order, with more than one pair of centres, distortions that are non-conjugate for one pair may be conjugate for another pair.