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XXX.—Non-Alternate ± Knots

Published online by Cambridge University Press:  06 July 2012

C. N. Little
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Extract

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, I understand by a knot a continuum which can not be brought to a circular form.

Type
Research Article
Information
Earth and Environmental Science Transactions of The Royal Society of Edinburgh , Volume 39 , Issue 3 , 1900 , pp. 771 - 778
Copyright
Copyright © Royal Society of Edinburgh 1900

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References

page 771 note * “Vorstudien zur Topologie,” Göttinger Studien, 1847, pp. 859–866. To the kind courtesy of Professors Felix Klein and P. Stäckel, 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:—

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.

page 771 note † 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.

page 771 note ‡ These Transactions, vol. xxxii. pp. 281–309; ibid., pp. 483–491.

page 771 note § Trans. Connecticut Academy, vol. vii. pp. 1–17; these Transactions, vol. xxxvi. pp. 253–255.

page 772 note * The classification of alternate knots according to the number of parts in the projection, in that set of vertically opposite compartments which has the smaller number, does not answer for non-alternates, since the same knot can be projected in forms belonging to different classes. Later, in § 9, a new basis of classification will be proposed. In this and the following sections, however, the term class has the old signification.

page 774 note * Trans. R.S.E., vol. xxxvi. pl. I. (39. D1. μ.)

page 775 note * Trans. R.S.E., vol. xxxii.; Trans. Conn. Acad., vol. vii.

page 776 note * By Dr H. F. Blichfeldt, now instructor, at the time a student in Stanford University, for whose care my thanks are due.