**KNOT**. 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 Listing. 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.. 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. I and 2. 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 has published contains a mere sketch of his investigations.

The most extensive dissertation on the properties of knots is that of Tait (*Trans. Roy. Soc. Edin.*, 1876-7). It was for the most part written in ignorance of the work of Listing, and was suggested by an inquiry concerning vortex atoms (see ATOM). 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, 1, 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, 3 and 4 and their perversions. Knots which can be deformed into their own perversion Tait calls " amphicheiral," 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. 5, 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," " beoriginal suggestion has been developed at considerable length by Boeddicker (*Erweiterung der Gauss'schen Theorie der Verschlingnungen*, Stuttgart, 1876). This author treats also of the connexion of knots with Riemann's surfaces.

It is to be noticed that, although every knot in which the crossings are alternately F*g- 6-over and under is irreducible, the converse is not generally true. This is obvious at once from fig. 6, which is merely the three-crossing knot with a doubled string— what Listing calls " paradromic."

Klein, in the *Mathematische Annalen* ix. 478, has proved the remarkable proposition that knots cannot exist in space of four dimensions.

**SAILORS' KNOTS**. - The knots used by sailors are of many kinds. The following are the most useful :- Overhand Knot (fig. 7). - Take the end a of the rope round the end b.

*Reef Knot* (figs. 8, 9). - Form an overhand knot as above. Then take the end a over the end b and through the bight. If the end a were taken under the end b a granny would be formed. This knot is so named from being used in tying the reef-points of a sail.

*Bowline* (figs. 10-12). - Lay the end a of a rope over the standing part b. Form with b a bight c over a. Take a round behind b and down through the bight c. This is a most useful knot employed to form a loop which will not slip.

*Half Hitch* (fig. 13). - Pass the end a of the rope round the standing part b and through the bight.

*Clove Hitch* (figs. 14, 15). - Pass the end a round a spar and cross it over b. l'ass it round the spar again and put the end a through the second bight.

*Blackwall Hitch*, (fig. 16). - Form a bight at the end of a rope, and put the hook of a tackle through the bight so that the end of the rope may he jammed between the standing part and the back of the hook.

*Timber Hitch* (fig. 17). - Take the end a of a rope round a spar, then round the standing part b, then several times round its own part c.

*Fisherman's Bend* (fig. 18). - Take two turns round a spar, then a half hitch round the standing part and between the spar and the turns, lastly a half hitch round the standing part.

*Carrick Bead* (fig. 19). - Lay the end of one rope over its OW11 standing part so as to form a bight. Put the cud of the other rope through this bight, under the standing part, over the end beyond the bight, under the standing part beyond the bight, and down through the bight over its own standing part.

*Sheet Bend* (fig. 20). - Pass the end of one rope through the bight of another, round. both parts of the other, and ender its own standing part.

*Single Wall Knot* (fig. 21). - .nifty the end of a rope, and with the strand a form a bight. Take the next strand b round the end. of a. Take the last strand c round the end of b and through the bight made by a. Haul the ends taut.

*Single Wall Crowned* (fig. 22). - Form a single wall, and lay one of the ends, a, over the knot. Lay b over a, and e over b and through the bight of a. Haul the ends taut.

*Double Wall and Double Crown* (fig. 23). - Form a single wall crowned ; then let the ends follow their ONVI1 parts round until all the parts appear double. Put the ends down through the knot.

*Matthew Walker* (figs. 24, 25). - Unlay the end of a rope. Take the first strand round the rope and through its own bight ; the second strand round the rope, through the bight of the first, and through its own bight ; the third through all three bights. Haul the ends taut.

See Bares, Seamanship, 4th ed., 1868; Dana, Seaman's Manual, 9th ed., 1863 ; A. H. Alston, Seamanship, Portsmouth, 1871 ; Kipping, Masting and Higging, 9th ed., 1864; Yachts and Yachting, by " Vanderdecken " (William Cooper), 1873 ; Book of Knots, by " Tom Bowling " (J. Bonwick), 1866.