bights a, a rove through the other and attached to the whip or
tackle.
For a complete treatise on the subject the reader may be referred to The Book of Knots, being a Complete Treatise on the Art of Cordage, illustrated by 172 Diagrams, showing the Manner of making every Knot, Tie and Splice, by Tom Bowling (London, 1890).
Mathematical Theory of Knots.
In the scientific sense a knot is an endless physical line which cannot be deformed into a circle. A physical line is flexible and inextensible, and cannot be cut—so that no lap of it can be drawn through another.
The founder of the theory of knots is undoubtedly Johann Benedict Listing (1808–1882). In his “Vorstudien zur Topologie” (Göttinger Studien, 1847), a work in many respects of startling originality, a few pages only are devoted to the subject.[1] He treats knots from the elementary notion of twisting one physical line (or thread) round another, and shows that from the projection of a knot on a surface we can thus obtain a notion of the relative situation of its coils. He distinguishes “reduced” from “reducible” forms, the number of crossings in the reduced knot being the smallest possible. The simplest form of reduced knot is of two species, as in figs. 49 and 50. Listing points out that these are formed, the first by right-handed the second by left-handed twisting. In fact, if three half-twists be given to a long strip of paper, and the ends be then pasted together, the two edges become one line, which is the knot in question. We may free it by slitting the paper along its middle line; and then we have the juggler’s trick of putting a knot on an endless unknotted band. One of the above forms cannot be deformed into the other. The one is, in Listing’s language, the “perversion” of the other, i.e. its image in a plane mirror. He gives a method of symbolizing reduced knots, but shows that in this method the same knot may, in certain cases, be represented by different symbols. It is clear that the brief notice he published contains a mere sketch of his investigations.
The most extensive dissertation on the properties of knots is that of Peter Guthrie Tait (Trans. Roy. Soc. Edin., xxviii. 145, where the substance of a number of papers in the Proceedings of the same society is reproduced). It was for the most part written in ignorance of the work of Listing, and was suggested by an inquiry concerning vortex atoms.
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| Fig. 49. | Fig. 50. | Fig. 51. | Fig. 52. |
Tait starts with the almost self-evident proposition that, if any plane closed curve have double points only, in passing continuously along the curve from one of these to the same again an even number of double points has been passed through. Hence the crossings may be taken alternately over and under. On this he bases a scheme for the representation of knots of every kind, and employs it to find all the distinct forms of knots which have, in their simplest projections, 3, 4, 5, 6 and 7 crossings only. Their numbers are shown to be 1, 1, 2, 4 and 8. The unique knot of three crossings has been already given as drawn by Listing. The unique knot of four crossings merits a few words, because its properties lead to a very singular conclusion. It can be deformed into any of the four forms—figs. 51 and 52 and their perversions. Knots which can be deformed into their own perversion Tait calls “amphicheiral” (from the Greek ἀμφί, on both sides, around, χείρ, hand), and he has shown that there is at least one knot of this kind for every even number of crossings. He shows also that “links” (in which two endless physical lines are linked together) possess a similar property; and he then points out that there is a third mode of making a complex figure of endless physical lines, without either knotting or linking. This may be called “lacing” or “locking.” Its nature is obvious from fig. 53, in which it will be seen that no one of the three lines is knotted, no two are linked, and yet the three are inseparably fastened together.
The rest of Tait’s paper deals chiefly with numerical characteristics of knots, such as their “knottiness,” “beknottedness” and “knotfulness.” He also shows that any knot, however complex, can be fully represented by three closed plane curves, none of which has double points and no two of which intersect. It may be stated here that the notion of beknottedness is founded on a remark of Gauss, who in 1833 considered the problem of the number of inter-linkings of two closed circuits, and expressed it by the electrodynamic measure of the work required to carry a unit magnetic pole round one of the interlinked curves, while a unit electric current is kept circulating in the other. This original suggestion has been developed at considerable length by Otto Boeddicker (Erweiterung der Gauss’schen Theorie der Verschlingungen (Stuttgart, 1876). This author treats also of the connexion of knots with Riemann’s surfaces.
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| Fig. 53. | Fig. 54. |
It is to be noticed that, although every knot in which the crossings are alternately over and under is irreducible, the converse is not generally true. This is obvious at once from fig. 54, which is merely the three-crossing knot with a doubled string—what Listing calls “paradromic.”
Christian Felix Klein, in the Mathematische Annalen, ix. 478, has proved the remarkable proposition that knots cannot exist in space of four dimensions. (P. G. T.)
KNOUT (from the French transliteration of a Russian word of Scandinavian origin; cf. A.-S. cnotta, Eng. knot), the whip used
in Russia for flogging criminals and political offenders. It is
said to have been introduced under Ivan III. (1462–1505). The knout had different forms. One was a lash of raw hide, 16 in.
long, attached to a wooden handle, 9 in. long. The lash ended
in a metal ring, to which was attached a second lash as
long, ending also in a ring, to which in turn was attached a few
inches of hard leather ending in a beak-like hook. Another kind
consisted of many thongs of skin plaited and interwoven with
wire, ending in loose wired ends, like the cat-o’-nine tails. The
victim was tied to a post or on a triangle of wood and stripped,
receiving the specified number of strokes on the back. A sentence
of 100 or 120 lashes was equivalent to a death sentence;
but few lived to receive so many. The executioner was usually
a criminal who had to pass through a probation and regular
training; being let off his own penalties in return for his services.
Peter the Great is traditionally accused of knouting his son
Alexis to death, and there is little doubt that the boy was
actually beaten till he died, whoever was the executioner. The
emperor Nicholas I. abolished the earlier forms of knout and
substituted the pleti, a three-thonged lash. Ostensibly the knout
has been abolished throughout Russia and reserved for the penal
settlements.
KNOWLES, SIR JAMES (1831–1908), English architect and editor, was born in London in 1831, and was educated, with a view to following his father’s profession, as an architect at University College and in Italy. His literary tastes also brought him at an early age into the field of authorship. In 1860 he published The Story of King Arthur. In 1867 he was introduced to Tennyson, whose house, Aldworth, on Blackdown, he designed; this led to a close friendship, Knowles assisting Tennyson in business matters, and among other things helping to design scenery for The Cup, when Irving produced that play in 1880. Knowles became intimate with a number of the most
interesting men of the day, and in 1869, with Tennyson’s co-operation, he started the Metaphysical Society, the object of which was to attempt some intellectual rapprochement between religion and science by getting the leading representatives of faith and unfaith to meet and exchange views.
The members from first to last were as follows: Dean Stanley, Seeley, Roden Noel, Martineau, W. B. Carpenter, Hinton, Huxley, Pritchard, Hutton, Ward, Bagehot, Froude, Tennyson, Tyndall, Alfred Barry, Lord Arthur Russell, Gladstone, Manning, Knowles, Lord Avebury, Dean Alford, Alex. Grant, Bishop Thirlwall, F. Harrison, Father Dalgairns, Sir G. Grove, Shadworth Hodgson,
- ↑ See P. G. Tait “On Listing’s Topologie,” Phil. Mag., xvii. 30.
