1.
Carrick bend
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The carrick bend is a knot used for joining two lines. It is particularly appropriate for very heavy rope or cable that is too large and it will not jam even after carrying a significant load or being soaked with water. As with many members of the basket weave knot family. The knot features prominently as a motif in the science fiction novel Picoverse by Robert A. Metzger. In heraldry, this known as the Wake knot or Ormonde knot and this knots name dates back to at least 1783, when it was included in a nautical bilingual dictionary authored by Daniel Lescallier. Its origins prior to that are not known with certainty, there are several possible explanations for the name Carrick being associated with this bend. The Elizabethan era plasterwork of Ormonde Castle in Carrick-on-Suir shows numerous carrick bends molded in relief, or the name may come from Carrick Roads—a large natural anchorage by Falmouth in Cornwall, England. The name may also have derived from the Carrack, a medieval type of ship. The eight crossings within the carrick bend allow for many similar-looking knots to be made, the lines in a full or true carrick bend alternate between over and under at every crossing. There are also two ways the ends can emerge from the knot, diagonally opposed or from the same side, the latter form is also called the double coin knot. The form with the ends emerging diagonally opposed is considered more secure, unfortunately, with so many permutations, the carrick bend is prone to being tied incorrectly. The carrick bend is generally tied in a flat interwoven form as shown above, without additional measures it will collapse into a different shape when tightened, a process known as capsizing, with the degree of capsizing depending on the looseness of the weave. This capsized form is both secure and stable once tightened, although it is bulkier than the form below. Incomplete capsizing resulting from a tight weave produces a form that is secure and stable. In the interest of making the carrick bend easier to untie, especially when tied in extremely large rope and this practice also keeps the knots profile flatter and can ease its passage over capstans or winches. The ends are traditionally seized to their standing part using a round seizing, for expediency, a series of double constrictor knots, drawn very tight, may also be used. When seizing the carrick bend, both ends must be secured to their parts or the bend will slip. In the decorative variation, both standing ends enter from one side and both working ends exit from the other, in this configuration the knot is known as the Josephine knot or double coin knot

2.
Turk's head
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A Turks head knot is a decorative knot with a variable number of interwoven strands, forming a closed loop. The name is used to describe the family of all such knots rather than one individual knot. While generally seen made around a cylinder, the knot can also be deformed into a flat, some variants can be arranged into a roughly spherical shape, akin to a monkeys fist knot. The knot is used primarily for decoration and occasionally as anti-chafing protection, a notable practical use for the Turks head is to mark the king spoke of a ships wheel, when this spoke is upright the rudder is in a central position. The knot takes its name from a resemblance to a turban. The Turks head knot is used as a woggle by Scout Leaders who completed their course and were thus awarded with the Wood Badge insignia. Each type of Turks head knot is classified according to the number of leads and bights, the number of bights is the number of crossings it makes as it goes around the circumference of the cylinder. The number of leads is the number of strands around the circumference of the cylinder, before doubling, tripling, depending on the number of leads and bights, a Turks head may be tied using a single strand or multiple strands. For example,3 lead ×5 bight, or 5 lead ×7 bight, there are three groupings of Turks head knots. The number of bights determines the shape found at the center, three bights create a triangular shape, while four create a square. A two lead,3 bight Turks head is an overhand knot. A two lead, three bight Turks head is also a trefoil knot if the ends are joined together, alternating torus knots are Turks head knots. The World Organization of scouting uses a variation of the Turks head knot called a woggle to affix their neckerchiefs and it is an official part of the uniform. How to tie a Turks head knot Shurdington Turks head Knot So-You-Want to Make a Rope Rug Eh

3.
Austrian knot
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An Austrian knot is an elaborate design of twisted cord or lace worn as part of a dress uniform, usually on the lower sleeve. It is usually a distinction worn by officers, the exception is the hussars. British cadet under officers wear Austrian knots as part of their rank insignia, while of Hungarian origin Vitéz kötés in English Bravery knot, the Austrian knot evolved as an indicator of rank among hussar officers of the Austrian Army in the 18th century. Epaulettes were widely perceived amongst the government in Vienna as a foreign influence, in the hussar regiments ranks came to be denoted by braided gold cords on the sleeve, with the number of gold cords representing the rank of the officer. Other branches of the Austrian Army used a system of waist-sashes, along with most other elaborate and conspicuous indicators of rank, Austrian knots fell into disuse during the First World War and were not revived in everyday wear. An exception was the French Army where the kepis still worn by most officers have Austrian knots in cruciform pattern on the top crown and they are still worn on some parade uniforms in France, where they are called noeuds hongrois. During the American Civil War, Confederate officers often wore gold Austrian knots on their uniforms, more elaborate braiding indicated higher rank. This type of insignia was worn by officers of the US Army on the sleeves of the full dress uniforms authorised until 1917. It is a feature of the mess dress uniform adopted as optional wear for officers in 1937

4.
The Ashley Book of Knots
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The Ashley Book of Knots is an encyclopedia of knots written and illustrated by the American artist Clifford W. Ashley. First published in 1944, it was the culmination of over 11 years of work, the book contains more than 3800 numbered entries and an estimated 7000 illustrations. The entries include knot instructions, uses, and some histories and it remains one of the most important and comprehensive books on knots. Due to its scope and wide availability The Ashley Book of Knots has become a significant reference work in the field of knotting, the numbers Ashley assigned to each knot can be used to unambiguously identify them. This helps to identify knots despite local colloquialisms or identification changes, citations to Ashley numbers are usually in the form, The Constrictor Knot, ABOK #1249 or even simply #1249 if the context of the reference is clear or already established. The book title is also found abbreviated in the forms, TABOK, some knots have more than one Ashley number due to having multiple uses or forms. For example, the entry for #1249 is in the chapter on binding knots. The commentary on some knots may fail to address their behavior when tied with synthetic fiber or kernmantle style ropes. Ashley suffered a stroke the year after his magnum opus was published. He was not able to produce an erratum or oversee a corrected edition, corrections submitted by the International Guild of Knot Tyers were incorporated in 1991. The original list of revisions submitted to the publisher is believed to have been lost, but many had been collected from a series of articles in Knotting Matters, additional errors have been identified since the 1991 corrections. At least one knot, the Hunters bend, was added in 1979, ISBN 0-385-04025-3 Reprint, Doubleday, New York 1963–1979, ISBN 0-571-09659-X

5.
Braid theory
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In topology, a branch of mathematics, braid theory is an abstract geometric theory studying the everyday braid concept, and some generalizations. The idea is that braids can be organized into groups, in which the operation is do the first braid on a set of strings. Such groups may be described by explicit presentations, as was shown by Emil Artin, for an elementary treatment along these lines, see the article on braid groups. Braid groups are also understood by a deeper mathematical interpretation, as the group of certain configuration spaces. To explain how to reduce a braid group in the sense of Artin to a fundamental group and that is, an ordered n-tuple is in the same orbit as any other that is a re-ordered version of it. A path in the symmetric product is the abstract way of discussing n points of X, considered as an unordered n-tuple. Since we must require that the strings never pass through other, it is necessary that we pass to the subspace Y of the symmetric product. That is, we remove all the subspaces of Xn defined by conditions xi = xj and this is invariant under the symmetric group, and Y is the quotient by the symmetric group of the non-excluded n-tuples. Under the dimension condition Y will be connected, with this definition, then, we can call the braid group of X with n strings the fundamental group of Y. The case where X is the Euclidean plane is the one of Artin. In some cases it can be shown that the homotopy groups of Y are trivial. When X is the plane, the braid can be closed, i. e. corresponding ends can be connected in pairs, to form a link, i. e. a possibly intertwined union of possibly knotted loops in three dimensions. The number of components of the link can be anything from 1 to n, a theorem of J. W. Alexander demonstrates that every link can be obtained in this way as the closure of a braid. Different braids can give rise to the link, just as different crossing diagrams can give rise to the same knot. Markov describes two moves on braid diagrams that yield equivalence in the corresponding closed braids, a single-move version of Markovs theorem, was published by Lambropoulou & Rourke. Vaughan Jones originally defined his polynomial as an invariant and then showed that it depended only on the class of the closed braid. The braid index is the least number of strings needed to make a closed braid representation of a link and it is equal to the least number of Seifert circles in any projection of a knot. Additionally, the length is the longest dimension of a braid

6.
Bridge number
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In the mathematical field of knot theory, the bridge number is an invariant of a knot defined as the minimal number of bridges required in all the possible bridge representations of a knot. Given a knot or link, draw a diagram of the using the convention that a gap in the line denotes an undercrossing. Call an arc in this diagram a bridge if it includes at least one overcrossing, then the bridge number of a knot can be found as the minimum number of bridges required for any diagram of the knot. Bridge number was first studied in the 1950s by Horst Schubert, the bridge number can equivalently be defined geometrically instead of topologically. In bridge representation, a knot lies entirely in the plane apart for a number of bridges whose projections onto the plane are straight lines. Equivalently the bridge number is the number of local maxima of the projection of the knot onto a vector. Every non-trivial knot has bridge number at least two, so the knots that minimize the number are the 2-bridge knots. It can be shown that every knot can be decomposed into two trivial n-tangles and hence 2-bridge knots are rational knots. If K is the sum of K1 and K2, then the bridge number of K is one less than the sum of the bridge numbers of K1. Crossing number Linking number Stick number Unknotting number Cromwell, Peter

7.
Crossing number (knot theory)
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In the mathematical area of knot theory, the crossing number of a knot is the smallest number of crossings of any diagram of the knot. By way of example, the unknot has crossing number zero, tables of prime knots are traditionally indexed by crossing number, with a subscript to indicate which particular knot out of those with this many crossings is meant. The listing goes 31,41,51,52,61 and this order has not changed significantly since P. G. Tait published a tabulation of knots in 1877. There has been little progress on understanding the behavior of crossing number under rudimentary operations on knots. A big open question if the crossing number is additive when taking knot sums. It is also expected that a satellite of a knot K should have larger crossing number than K, additivity of crossing number under knot sum has been proven for special cases, for example if the summands are alternating knots, or if the summands are torus knots. Marc Lackenby has also given a proof that there is a constant N >1 such that 1 N ≤ c r, but his method, there are connections between the crossing number of a knot and the physical behavior of DNA knots. For prime DNA knots, crossing number is a predictor of the relative velocity of the DNA knot in agarose gel electrophoresis. Basically, the higher the number, the faster the relative velocity. For composite knots, this not appear to be the case. There are related concepts of average crossing number and asymptotic crossing number, both of these quantities bound the standard crossing number. Asymptotic crossing number is conjectured to be equal to crossing number, other numerical knot invariants include the bridge number, linking number, stick number, and unknotting number

8.
Seifert surface
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In mathematics, a Seifert surface is a surface whose boundary is a given knot or link. Such surfaces can be used to study the properties of the knot or link. For example, many knot invariants are most easily calculated using a Seifert surface, Seifert surfaces are also interesting in their own right, and the subject of considerable research. Specifically, let L be an oriented knot or link in Euclidean 3-space. Note that any compact, connected, oriented surface with nonempty boundary in Euclidean 3-space is the Seifert surface associated to its boundary link, a single knot or link can have many different inequivalent Seifert surfaces. A Seifert surface must be oriented and it is possible to associate surfaces to knots which are not oriented nor orientable, as well. The standard Möbius strip has the unknot for a boundary but is not considered to be a Seifert surface for the unknot because it is not orientable, the checkerboard coloring of the usual minimal crossing projection of the trefoil knot gives a Mobius strip with three half twists. As with the example, this is not a Seifert surface as it is not orientable. Applying Seiferts algorithm to this diagram, as expected, does produce a Seifert surface, in case, it is a punctured torus of genus g=1. It is a theorem that any link always has an associated Seifert surface and this theorem was first published by Frankl and Pontrjagin in 1930. A different proof was published in 1934 by Herbert Seifert and relies on what is now called the Seifert algorithm, the algorithm produces a Seifert surface S, given a projection of the knot or link in question. Suppose that link has m components, the diagram has d crossing points, then the surface S is constructed from f disjoint disks by attaching d bands. The homology group H1 is free abelian on 2g generators, the intersection form Q on H1 is skew-symmetric, and there is a basis of 2g cycles a1, a2. a2g with Q= the direct sum of g copies of. The 2g × 2g integer Seifert matrix V= has v the linking number in Euclidean 3-space of ai, every integer 2g × 2g matrix V with V − V * = Q arises as the Seifert matrix of a knot with genus g Seifert surface. The Alexander polynomial is computed from the Seifert matrix by A = d e t, the Alexander polynomial is independent of the choice of Seifert surface S, and is an invariant of the knot or link. The signature of a knot is the signature of the symmetric Seifert matrix V + V ⊤ and it is again an invariant of the knot or link. The genus of a knot K is the knot invariant defined by the genus g of a Seifert surface for K. For instance, An unknot—which is, by definition, the boundary of a genus zero