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.
| 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 electro-dynamic 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.
| 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.)
[1] See P. G. Tait “On Listing’s Topologie,” Phil. Mag., xvii. 30.