1.
List of decorative knots
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A decorative or ornamental knot is an often complex knot exhibiting repeating patterns. A decorative knot is generally a knot that not only has practical use but is known for its aesthetic qualities. Often originating from maritime use, decorative knots are not only serviceable and functional, decorative knots may be used alone or in combination, and may consist of single or multiple strands. Though the word decorative sometimes implies that little or no function is served and this is an alphabetical list of decorative knots. Coxcombing is a decorative knotwork performed by sailors, the general purpose to dress-up items and parts of ships and boats during the age of sail. Modern uses are to wrap boat tillers and ships wheels with small diameter line to enhance the grip as well as the nautical appeal, knots used in coxcombing include Turks head knot, Flemish, French whipping, and others

2.
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

3.
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

4.
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

5.
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

6.
Braid length
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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

7.
Braid number
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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

8.
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

9.
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

10.
Knot genus
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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

11.
Hyperbolic volume
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In the mathematical field of knot theory, the hyperbolic volume of a hyperbolic link is the volume of the links complement with respect to its complete hyperbolic metric. The volume is necessarily a finite number, and is a topological invariant of the link. As a link invariant, it was first studied by William Thurston in connection with his geometrization conjecture, the components of the link will become cusps of the 3-manifold, and the manifold itself will have finite volume. In particular, the volume of the complement is a knot invariant. In order to make it well-defined for all knots or links, there are only finitely many hyperbolic knots for any given volume. In practice, hyperbolic volume has proven effective in distinguishing knots. Jeffrey Weekss computer program SnapPea is the tool used to compute hyperbolic volume of a link. The Weeks manifold has the smallest possible volume of any closed manifold, Thurston and Jørgensen proved that the set of real numbers that are hyperbolic volumes of 3-manifolds is well-ordered, with order type ωω

12.
Unknotting number
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In the mathematical area of knot theory, the unknotting number of a knot is the minimum number of times the knot must be passed through itself to untie it. If a knot has unknotting number n, then there exists a diagram of the knot which can be changed to unknot by switching n crossings, the unknotting number of a knot is always less than half of its crossing number. Any composite knot has unknotting number at least two, and therefore every knot with unknotting number one is a prime knot. The following table show the numbers for the first few knots, In general. Known cases include, The unknotting number of a nontrivial twist knot is equal to one. The unknotting number of a knot is equal to /2. The unknotting numbers of knots with nine or fewer crossings have all been determined. Crossing number Bridge number Linking number Stick number Unknotting problem Three_Dimensional_Invariants#Unknotting_Number, The Knot Atlas

13.
Conway notation (knot theory)
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In knot theory, Conway notation, invented by John Horton Conway, is a way of describing knots that makes many of their properties clear. It composes a knot using certain operations on tangles to construct it, in Conway notation, the tangles are generally algebraic 2-tangles. This means their tangle diagrams consist of 2 arcs and 4 points on the edge of the diagram, furthermore, tangles consisting only of positive crossings are denoted by the number of crossings, or if there are only negative crossings it is denoted by a negative number. If the arcs are not crossed, or can be transformed to into a position with the Reidemeister moves, it is called the 0 or ∞ tangle. If a tangle, a, is reflected on the NW-SE line, tangles have three binary operations, sum, product, and ramification, however all can be explained using tangle addition and negation. The tangle product, a b, is equivalent to −a+b. rational tangles are equivalent if and only if their fractions are equal. An accessible proof of this fact is given in, a number before an asterisk, *, denotes the polyhedron number, multiple asterisks indicate that multiple polyhedra of that number exist. Dowker notation Alexander–Briggs notation Conway, J. H, an Enumeration of Knots and Links, and Some of Their Algebraic Properties. In J. Leech, Computational Problems in Abstract Algebra, pdf available online Louis H. Kauffman, Sofia Lambropoulou, On the classification of rational tangles. Advances in Applied Mathematics,33, No

14.
Knot theory
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In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, in mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3. Knots can be described in various ways, given a method of description, however, there may be more than one description that represents the same knot. For example, a method of describing a knot is a planar diagram called a knot diagram. Any given knot can be drawn in different ways using a knot diagram. Therefore, a problem in knot theory is determining when two descriptions represent the same knot. A complete algorithmic solution to this exists, which has unknown complexity. In practice, knots are often distinguished by using a knot invariant, important invariants include knot polynomials, knot groups, and hyperbolic invariants. The original motivation for the founders of theory was to create a table of knots and links. Over six billion knots and links have been tabulated since the beginnings of knot theory in the 19th century, to gain further insight, mathematicians have generalized the knot concept in several ways. Knots can be considered in other spaces and objects other than circles can be used. Higher-dimensional knots are n-dimensional spheres in m-dimensional Euclidean space, archaeologists have discovered that knot tying dates back to prehistoric times. Besides their uses such as recording information and tying objects together, knots have interested humans for their aesthetics, Knots appear in various forms of Chinese artwork dating from several centuries BC. The endless knot appears in Tibetan Buddhism, while the Borromean rings have made repeated appearances in different cultures, the Celtic monks who created the Book of Kells lavished entire pages with intricate Celtic knotwork. Mathematical studies of knots began in the 19th century with Gauss, in the 1860s, Lord Kelvins theory that atoms were knots in the aether led to Peter Guthrie Taits creation of the first knot tables for complete classification. Tait, in 1885, published a table of knots with up to ten crossings and this record motivated the early knot theorists, but knot theory eventually became part of the emerging subject of topology. This would be the approach to knot theory until a series of breakthroughs transformed the subject. In the late 1970s, William Thurston introduced hyperbolic geometry into the study of knots with the hyperbolization theorem, many knots were shown to be hyperbolic knots, enabling the use of geometry in defining new, powerful knot invariants

15.
Dowker notation
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In the mathematical field of knot theory, the Dowker notation, also called the Dowker–Thistlethwaite notation or code, for a knot is a sequence of even integers. The notation is named after Clifford Hugh Dowker and Morwen Thistlethwaite, to generate the Dowker notation, traverse the knot using an arbitrary starting point and direction. Label each of the n crossings with the numbers 1, 2n in order of traversal, with the following modification, if the label is an even number and the strand followed crosses over at the crossing, then change the sign on the label to be a negative. When finished, each crossing will be labelled a pair of integers, one even, the Dowker notation is the sequence of even integer labels associated with the labels 1,3. For example, a diagram may have crossings labelled with the pairs. The Dowker notation for this labelling is the sequence,6 −1228 −4 −10, Knots tabulations typically consider only prime knots and disregard chirality, so this ambiguity does not affect the tabulation. The ménage problem, posed by Tait, concerns counting the number of different number sequences possible in this notation, conway notation Alexander–Briggs notation Adams, Colin Conrad. The Knot Book, An Elementary Introduction to the Mathematical Theory of Knots

16.
8 19 knot
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In knot theory, a torus knot is a special kind of knot that lies on the surface of an unknotted torus in R3. Similarly, a link is a link which lies on the surface of a torus in the same way. Each torus knot is specified by a pair of coprime integers p and q, a torus link arises if p and q are not coprime. A torus knot is trivial if and only if p or q is equal to 1 or −1. The simplest nontrivial example is the knot, also known as the trefoil knot. A torus knot can be rendered geometrically in multiple ways which are topologically equivalent, the convention used in this article and its figures is the following. The -torus knot winds q times around a circle in the interior of the torus, if p and q are not relatively prime, then we have a torus link with more than one component. The direction in which the strands of the wrap around the torus is also subject to differing conventions. The most common is to have the form a right-handed screw for p q >0. The -torus knot can be given by the parametrization x = r cos y = r sin z = − sin where r = cos +2 and 0 < ϕ <2 π. This lies on the surface of the torus given by 2 + z 2 =1, other parameterizations are also possible, because knots are defined up to continuous deformation. The latter generalizes smoothly to any coprime p, q satisfying p < q <2 p, a torus knot is trivial iff either p or q is equal to 1 or −1. Each nontrivial torus knot is prime and chiral, the torus knot is equivalent to the torus knot. This can be proved by moving the strands on the surface of the torus, the torus knot is the obverse of the torus knot. The torus knot is equivalent to the torus knot except for the reversed orientation, any -torus knot can be made from a closed braid with p strands. The appropriate braid word is q, the crossing number of a torus knot with p, q >0 is given by c = min. The genus of a knot with p, q >0 is g =12. The Alexander polynomial of a knot is

17.
Alternating knot
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In knot theory, a knot or link diagram is alternating if the crossings alternate under, over, under, over, as one travels along each component of the link. A link is alternating if it has an alternating diagram, many of the knots with crossing number less than 10 are alternating. The simplest non-alternating prime knots have 8 crossings and it is conjectured that as the crossing number increases, the percentage of knots that are alternating goes to 0 exponentially quickly. Alternating links end up having an important role in theory and 3-manifold theory, due to their complements having useful and interesting geometric. This led Ralph Fox to ask, What is an alternating knot, by this he was asking what non-diagrammatic properties of the knot complement would characterize alternating knots. Various geometric and topological information is revealed in an alternating diagram, primeness and splittability of a link is easily seen from the diagram. The crossing number of a reduced, alternating diagram is the number of the knot. This last is one of the celebrated Tait conjectures, an alternating knot diagram is in one-to-one correspondence with a planar graph. Each crossing is associated with an edge and half of the components of the complement of the diagram are associated with vertices in a checker board manner. The Tait conjectures are, Any reduced diagram of a link has the fewest possible crossings. Any two reduced diagrams of the alternating knot have the same writhe. Given any two reduced alternating diagrams D1 and D2 of an oriented, prime alternating link, D1 may be transformed to D2 by means of a sequence of certain simple moves called flypes, also known as the Tait flyping conjecture. Morwen Thistlethwaite, Louis Kauffman and K. Murasugi proved the first two Tait conjectures in 1987 and Morwen Thistlethwaite and William Menasco proved the Tait flyping conjecture in 1991, thus hyperbolic volume is an invariant of many alternating links. Marc Lackenby has shown that the volume has upper and lower linear bounds as functions of the number of twist regions of a reduced, adams, The Knot Book, An elementary introduction to the mathematical theory of knots. American Mathematical Society, Providence, RI,2004, xiv+307 pp. ISBN 0-8218-3678-1 William Menasco, Closed incompressible surfaces in alternating knot and link complements. Marc Lackenby, The volume of hyperbolic alternating link complements, with an appendix by Ian Agol and Dylan Thurston. Weisstein, Eric W. Taits Knot Conjectures, celtic Knotwork to build an alternating knot from its planar graph

18.
Fibered knot
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For example, The unknot, trefoil knot, and figure-eight knot are fibered knots. The Hopf link is a fibered link, fibered knots and links arise naturally, but not exclusively, in complex algebraic geometry. For instance, each point of a complex plane curve can be described topologically as the cone on a fibered knot or link called the link of the singularity. The trefoil knot is the link of the cusp singularity z 2 + w 3, in these cases, the family of Seifert surfaces is an aspect of the Milnor fibration of the singularity. A knot is fibered if and only if it is the binding of some open book decomposition of S3, the Alexander polynomial of a fibered knot is monic, i. e. the coefficients of the highest and lowest powers of t are plus or minus 1. Examples of knots with nonmonic Alexander polynomials abound, for example the twist knots have Alexander polynomials qt − + qt−1, in particular the Stevedores knot is not fibered. Pretzel knot How to construct all fibered knots and links

19.
Prime knot
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In knot theory, a prime knot or prime link is a knot that is, in a certain sense, indecomposable. Specifically, it is a knot which cannot be written as the knot sum of two non-trivial knots. Knots that are not prime are said to be composite knots or composite links and it can be a nontrivial problem to determine whether a given knot is prime or not. A family of examples of prime knots are the torus knots and these are formed by wrapping a circle around a torus p times in one direction and q times in the other, where p and q are coprime integers. The simplest prime knot is the trefoil with three crossings, the trefoil is actually a -torus knot. The figure-eight knot, with four crossings, is the simplest non-torus knot, for any positive integer n, there are a finite number of prime knots with n crossings. The first few values are given in the following table, enantiomorphs are counted only once in this table and the following chart. A theorem due to Horst Schubert states that every knot can be expressed as a connected sum of prime knots. List of prime knots Weisstein, Eric W, prime Links with a Non-Prime Component, The Knot Atlas

20.
Fully amphichiral knot
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In the mathematical field of knot theory, a chiral knot is a knot that is not equivalent to its mirror image. An oriented knot that is equivalent to its image is an amphichiral knot. The chirality of a knot is a knot invariant, a knots chirality can be further classified depending on whether or not it is invertible. The chirality of certain knots was long suspected, and was proven by Max Dehn in 1914, P. G. Tait conjectured that all amphichiral knots had even crossing number, but a counterexample was found by Morwen Thistlethwaite et al. in 1998. However, Taits conjecture was true for prime, alternating knots. The simplest chiral knot is the knot, which was shown to be chiral by Max Dehn. The Alexander polynomial cannot detect the chirality of a knot, but the Jones polynomial can in some cases, if Vk ≠ Vk, then the knot is chiral, however the converse is not true. The HOMFLY polynomial is even better at detecting chirality, but there is no known polynomial knot invariant which can fully detect chirality, a chiral knot that is invertible is classified as a reversible knot. If a knot is not equivalent to its inverse or its image, it is a fully chiral knot. An amphichiral knot is one which has an orientation-reversing self-homeomorphism of the 3-sphere, α, all amphichiral alternating knots have even crossing number. The first amphichiral knot with odd crossing number is a 15-crossing knot discovered by Hoste et al, if a knot is isotopic to both its reverse and its mirror image, it is fully amphichiral. The simplest knot with this property is the figure-eight knot, if the self-homeomorphism, α, preserves the orientation of the knot, it is said to be positive amphichiral. This is equivalent to the knot being isotopic to its mirror, no knots with crossing number smaller than twelve are positive amphichiral. If the self-homeomorphism, α, reverses the orientation of the knot and this is equivalent to the knot being isotopic to the reverse of its mirror image. The knot with this property that has the fewest crossings is the knot 817

21.
Knot
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A knot is a method of fastening or securing linear material such as rope by tying or interweaving. It may consist of a length of one or several segments of rope, string, webbing, twine, strap, Knots have been the subject of interest for their ancient origins, their common uses, and the area of mathematics known as knot theory. There is a variety of knots, each with properties that make it suitable for a range of tasks. Some knots are used to attach the rope to other such as another rope, cleat, ring. Some knots are used to bind or constrict objects, decorative knots usually bind to themselves to produce attractive patterns. While some people can look at diagrams or photos and tie the illustrated knots, Knot tying skills are often transmitted by sailors, scouts, climbers, canyoners, cavers, arborists, rescue professionals, stagehands, fishermen, linemen and surgeons. The International Guild of Knot Tyers is a dedicated to the Promotion of Knot tying. Truckers in need of securing a load may use a truckers hitch, Knots can save spelunkers from being buried under rock. Many knots can also be used as tools, for example, the bowline can be used as a rescue loop. The diamond hitch was used to tie packages on to donkeys. In hazardous environments such as mountains, knots are very important, note the systems mentioned typically require carabineers and the use of multiple appropriate knots. These knots include the bowline, double figure eight, munter hitch, munter mule, prusik, autoblock, thus any individual who goes into a mountainous environment should have basic knowledge of knots and knot systems to increase safety and the ability to undertake activities such as rappelling. Knots can be applied in combination to produce objects such as lanyards. In ropework, the end of a rope is held together by a type of knot called a whipping knot. Many types of textiles use knots to repair damage, macrame, one kind of textile, is generated exclusively through the use of knotting, instead of knits, crochets, weaves or felting. Macramé can produce self-supporting three-dimensional textile structures, as well as flat work, Knots weaken the rope in which they are made. When knotted rope is strained to its point, it almost always fails at the knot or close to it. The bending, crushing, and chafing forces that hold a knot in place also unevenly stress rope fibers, the exact mechanisms that cause the weakening and failure are complex and are the subject of continued study

22.
Endless knot
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The endless knot or eternal knot is a symbolic knot and one of the Eight Auspicious Symbols. It is an important cultural marker in places significantly influenced by Tibetan Buddhism such as Tibet, Mongolia, Tuva, Kalmykia and it is also sometimes found in Chinese art and used in Chinese knots. The endless knot has been described as an ancient symbol representing the interweaving of the Spiritual path, all existence, it says, is bound by time and change, yet ultimately rests serenely within the Divine and the Eternal. Various interpretations of the symbol are, The eternal continuum of mind, the endless knot iconography symbolised Samsara i. e. the endless cycle of suffering or birth, death and rebirth within Tibetan Buddhism. The inter-twining of wisdom and compassion, interplay and interaction of the opposing forces in the dualistic world of manifestation, leading to their union, and ultimately to harmony in the universe. The mutual dependence of religious doctrine and secular affairs, the union of wisdom and method. The inseparability of emptiness and dependent origination, the reality of existence. Symbolic of knot symbolism in linking ancestors and omnipresence Since the knot has no beginning or end it also symbolizes the wisdom of the Buddha. See 7₄ knot for decorations or symbols in other cultures which are equivalent to the interlaced form of the simplest version of the Buddhist endless knot

23.
Bight (knot)
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In knot tying, a bight is a curved section or slack part between the two ends of a rope, string, or yarn. Any section of line that is bent into a U-shape is a bight, an open loop is a curve in a rope narrower than a bight but with separated ends. The term is used in a more specific way when describing Turks head knots. In order to make a knot, a bight must be passed. This slipped form of the knot is more easily untied, the traditional bow knot used for tying shoelaces is simply a reef knot with the final overhand knot made with two bights instead of the ends. Similarly, a hitch is a slipped clove hitch. The phrase in the means a bight of line is itself being used to make a knot. Specifically this means that the knot can be formed without access to the ends of the rope. This can be an important property for knots to be used in situations where the ends of the rope are inaccessible, many knots normally tied with an end also have a form which is tied in the bight. In other cases, a knot being tied in the bight is a matter of the method of tying rather than a difference in the form of the knot. For example, the hitch can be made in the bight if it is being slipped over the end of a post but not if being cast onto a closed ring. Other knots, such as the knot, cannot be tied in the bight without changing their final form

24.
Turk's head knot
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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

25.
Logo
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A logo is a graphic mark, emblem, or symbol commonly used by commercial enterprises, organizations, and even individuals to aid and promote instant public recognition. There are purely graphic emblems, symbols, icons and logos, in the days of hot metal typesetting, a logotype was one word cast as a single piece of type. By extension, the term was used for a uniquely set. At the level of communication and in common usage, a companys logo is today often synonymous with its trademark or brand. The arts were expanding in purpose—from expression and decoration of an artistic, storytelling nature, to a differentiation of brands, consultancies and trades-groups in the commercial arts were growing and organizing, by 1890, the US had 700 lithographic printing firms employing more than 8,000 people. Artistic credit tended to be assigned to the company, as opposed to the individual artists who usually performed less important jobs. Playful children’s books, authoritative newspapers, and conversational periodicals developed their own visual and editorial styles for unique, as printing costs decreased, literacy rates increased, and visual styles changed, the Victorian decorative arts led to an expansion of typographic styles and methods of representing businesses. A renewal of interest in craftsmanship and quality also provided the artists and companies with a greater interest in credit, leading to the creation of unique logos and marks. By the 1950s, Modernism had shed its roots as an artistic movement in Europe to become an international, commercialized movement with adherents in the United States. Modernist-inspired logos proved successful in the era of mass visual communication ushered in by television, improvements in printing technology, the current era of logo design began in the 1870s with the first abstract logo, the Bass red triangle. As of 2014, many corporations, products, brands, services, agencies, as a result, only a few of the thousands of ideograms in circulation are recognizable without a name. Ideograms and symbols may be effective than written names, especially for logos translated into many alphabets in increasingly globalized markets. For instance, a written in Arabic script might have little resonance in most European markets. By contrast, ideograms keep the general nature of a product in both markets. In non-profit areas, the Red Cross exemplifies a well-known emblem that does not need an accompanying name, the red cross and red crescent are among the best-recognized symbols in the world. National Red Cross and Red Crescent Societies and their Federation as well as the International Committee of the Red Cross include these symbols in their logos, branding can aim to facilitate cross-language marketing. Consumers and potential consumers can identify the Coca-Cola name written in different alphabets because of the standard color, the text was written in Spencerian Script, which was a popular writing style when the Coca Cola Logo was being designed. Since a logo is the visual entity signifying an organization, logo design is an important area of graphic design, a logo is the central element of a complex identification system that must be functionally extended to all communications of an organization

26.
International Guild of Knot Tyers
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The International Guild of Knot Tyers is a worldwide association for people with an interest in knots and knot tying. Des Pawson was a manager for a large stationery firm based in Ipswich. Geoffrey Budworth was a Metropolitan Police Inspector and knotting consultant, Des first wrote to Geoff on the 8th October 1978. Even then,1981 went by without further development, and this is a source of regret to them both as it was the centenary of Clifford W. Ashleys birth. The object of the Guild shall be the advancement of education by the study of and practice of the art, craft and science of knotting, past and present. In furtherance of this object but not otherwise the Guild shall have the powers, To undertake research into all aspects of knotting. To establish a body for consultation purposes To publish a periodical or periodicals and other papers and books about knotcrafts. To form a collection of knots and knotting and work related crafts, unlike a traditional guild no level of expertise is required for membership, only an interest in knotting. Members of the Guild assisted with revisions and corrections to The Ashley Book of Knots in 1991, knotting Matters is the quarterly news letter of the IGKT and is sent by post to all subscribed members. The first issue was published in Autumn 1982 and was 17 Pages long and in Black and white, the centennial was produced in September 2008 edited by Lindsey Philpott and was professionally printed with colour covers and was 50 pages in length. Knotting Matters is made from Guild members submissions and other news from the guild, the Guild dates from an inaugural meeting of 25 individuals aboard the Maritime Trusts vessel R. R. S. Discovery berthed in St. Katharines Dock in the lee of Tower Bridge London on April,1982. Those in attendance were Dr. Harry ASHER, Mr. Roy E. BAIL, Mr. C. G. BELLINGHAM, Mr. Geoffrey BUDWORTH, Mr. John CONSTABLE, Mr. Bernard J. CUTBUSH, Mrs. Anne DEVINE, Mr. Ron W. EVANS, Mr. Sid EVANS, Mr. Eric FRANKLIN, Mr. Frank HARRIS, Mr. John HAWES, Mr. Paul HERBERT, Dr. Edward HUNTER, Miss. Albert KIRBY, Mr. Allan McDOWALL, Mr. Desmond MANDEVILLE, Mr. Graham MOTT, Mr. Des PAWSON, Mrs. Liz PAWSON, Mr. Douglas PROBERT, Mr. W. Ettrick THOMSON, Mr. Don WOODS and Mr. Quinton WINCH. Mr. Robert CHISNALL of Kingston, Ontario, Canada, THOMASON of Queensland, Australia Both expressed a wish to be involved from the outset but due to distance were unable to attend the opening meeting. This involves tying six basic knots - reef knot, sheet bend, sheepshank, clove hitch, round turn, the authenticated world record is 8.1 seconds, set by Clinton R. Bailey, Sr. in 1977. IGKT members have discussed proposals for formal rules to govern future attempts on this record

27.
Woggle
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A woggle is a device to fasten the neckerchief, or scarf, worn as part of the Scout or Girl Guides uniform, originated by a Tasmanian Scout in the 1920s. Early Scouts tied a knot in their neckerchief to fasten it around the neck, in the United States, experiments were made with rings made from bone, rope or wood. The result was the Gilwell Woggle, on the origin of the Woggle, Shankley said, They used to knot their scarves, which used to get creased and stick out at the ends. But in America the early scouts used to plait up various stuffs to make a ring for theirs — they called it a boon-doggle, I got some thin sewing machine leather belting, plaited it into a neat ring, submitted it, and had it accepted. I called it a Woggle and that’s the name it’s known by throughout the world The earliest known reference to a Woggle is the June 1923 edition of The Scout. The term was applied to other designs of fastener, of many shapes and sizes. The Woggle designed by Bill Shankley became known as the Gillwell Woggle, trained leaders are admitted into 1st Gillwell Park Scout Troop, with the Gilwell Woggle as one of its symbols. Because of its association it is not worn by other scouts, at the 1989 US National Scout Jamboree, William Green Bar Bill said to Jim Newell, Francis Gidney was the Camp Chief for the first two courses at Gilwell Park. Francis Gidney knew that most folks were not good wood carvers and this in no way is intended to contradict the contribution made by Bill Shankley. Keas, Cubs, Scouts, Venturers and Rovers all wear either a standard woggle for their section, until trained to the Gilwell woggle level, leaders wear a plaited leather woggle with a dome fastening. One story relating to the origin of the word woggle is that it was named to rhyme with the word boon doggle used in America, however the term woggle pre-dates the first known reference to this in 1925. There are a few references to the word woggle before its adoption by the Scout movement. It is thought that woggle was a verb, with similar meanings to waggle and wobble and it was in limited use as a noun around 1900. In the US, the used to secure the neckerchief is called a neckerchief slide. An early photographic reference to a SLIDE is in the BSA magazine Scouting of 1 April 1917, the cover for November 1917 issue prominently shows a scout wearing a slide to hold the neckerchief in place. In the BSA magazine Scouting from August 1923 Page 7, the term slip-on, there is an example of a rams head made of bone and an illustration on how to make your own Turks head slip-on. The article also comments that the neckerchief should be tied using the four-in-hand knot when not using a slide, two months later, Boys Life magazine repeated many of the article key points. The article makes reference to making your own troop or patrol slide

28.
Loop (knot)
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This page explains commonly used terms related to knots. A bend is a used to join two lengths of rope. A bight has two meanings in knotting and it can mean either any central part of a rope or an arc in a rope that is at least as wide as a semicircle. In either case, a bight is a length of rope that does not cross itself, knots that can be tied without use of the working end are called knots on the bight. Binding knots are knots that either constrict a single object or hold two objects snugly together, whippings, seizings and lashings serve a similar purpose to binding knots, but contain too many wraps to be properly called a knot. In binding knots, the ends of rope are either joined together or tucked under the turns of the knot, another term for the working end. A knot that has capsized has deformed into a different structure, although capsizing is sometimes the result of incorrect tying or misuse, it can also be done purposefully in certain cases to strengthen the knot. Chirality is the handedness of a knot, topologically speaking, a knot and its mirror image may or may not have knot equivalence. A decorative knot is any aesthetically pleasing knot, although it is not necessarily the case, most decorative knots also have practical applications or were derived from other well-known knots. Decorative knotting is one of the oldest and most widely distributed folk art, knot dressing is the process of arranging a knot in such a way as to improve its performance. Crossing or uncrossing the rope in a way, depending on the knot. An elbow refers to any two nearby crossings of a rope, an elbow is created when an additional twist is made in a loop. A flake refers to any number of turns in a coiled rope, likewise, to flake a rope means to coil it. A friction hitch is a knot that attaches one rope to another in a way allows the knots position to easily be adjusted. Sometimes friction hitches are called slide-and-grip knots and they are often used in climbing applications. A hitch is a knot that attaches a rope to some object, often a ring, rail, spar, a jamming knot is any knot that becomes very difficult to untie after use. Knots that are resistant to jamming are called non-jamming knots, a lashing is an arrangement of rope used to secure two or more items together in a rigid manner. Common uses include the joining scaffolding poles and the securing of sailing masts, the square lashing, diagonal lashing, and shear lashing are well-known lashings used to bind poles perpendicularly, diagonally, and in parallel, respectively

29.
Turn (knot)
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A turn is one round of rope on a pin or cleat, or one round of a coil. Turns can be made around various objects, through rings, or around the part of the rope itself or another rope. A turn also denotes a component of a knot, when the legs of a loop are brought together and crossed the rope has taken a turn. One distinguishes between single turn, round turn, and two round turns depending on the number of revolutions around an object, the benefit of round turns is best understood from the capstan equation. A riding turn is a section of rope that passes on top of another section of rope, examples of riding turns can be seen in both the constrictor knot and the strangle knot. The second course of wrappings in some seizing knots can be referred to as riding turns, the formation of an unintentional riding turn on a sailing winch can cause it to jam. A single hitch is a type of knot and this hitch is actually a turn tied around an object where the end is secured by its own standing part

30.
List of knot terminology
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This page explains commonly used terms related to knots. A bend is a used to join two lengths of rope. A bight has two meanings in knotting and it can mean either any central part of a rope or an arc in a rope that is at least as wide as a semicircle. In either case, a bight is a length of rope that does not cross itself, knots that can be tied without use of the working end are called knots on the bight. Binding knots are knots that either constrict a single object or hold two objects snugly together, whippings, seizings and lashings serve a similar purpose to binding knots, but contain too many wraps to be properly called a knot. In binding knots, the ends of rope are either joined together or tucked under the turns of the knot, another term for the working end. A knot that has capsized has deformed into a different structure, although capsizing is sometimes the result of incorrect tying or misuse, it can also be done purposefully in certain cases to strengthen the knot. Chirality is the handedness of a knot, topologically speaking, a knot and its mirror image may or may not have knot equivalence. A decorative knot is any aesthetically pleasing knot, although it is not necessarily the case, most decorative knots also have practical applications or were derived from other well-known knots. Decorative knotting is one of the oldest and most widely distributed folk art, knot dressing is the process of arranging a knot in such a way as to improve its performance. Crossing or uncrossing the rope in a way, depending on the knot. An elbow refers to any two nearby crossings of a rope, an elbow is created when an additional twist is made in a loop. A flake refers to any number of turns in a coiled rope, likewise, to flake a rope means to coil it. A friction hitch is a knot that attaches one rope to another in a way allows the knots position to easily be adjusted. Sometimes friction hitches are called slide-and-grip knots and they are often used in climbing applications. A hitch is a knot that attaches a rope to some object, often a ring, rail, spar, a jamming knot is any knot that becomes very difficult to untie after use. Knots that are resistant to jamming are called non-jamming knots, a lashing is an arrangement of rope used to secure two or more items together in a rigid manner. Common uses include the joining scaffolding poles and the securing of sailing masts, the square lashing, diagonal lashing, and shear lashing are well-known lashings used to bind poles perpendicularly, diagonally, and in parallel, respectively

31.
Albright special
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The Albright special or Albright knot is a knot used in angling. It is a knot used to tie two different diameters of line together, for instance to tie monofilament to braid. The Albright is relatively smooth and passes through guides when required, some anglers coat the knot with a rubber based cement to make it even smoother and more secure. When tying, it is important to wind the loops neatly around the loop of larger line, List of bend knots List of knots Animated Albright knot video and Step by Step Procedure Video instructions on how to tie an Albright knot Grog

32.
Ashley's bend
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Ashleys bend is a knot used to securely join the ends of two ropes together. It is similar to several related bend knots which consist of two interlocking overhand knots, and in particular the alpine butterfly bend and these related bends differ by the way the two constituent overhand knots are interlocked. The name Ashleys bend is now associated with the knot described in entry #1452 of The Ashley Book of Knots, clifford Ashley developed this bend and believed it to be original, along with several similar ones. Rather than giving it a name he simply noted the date when he first tied it. Cyrus L. Day, a contemporary of Ashleys, called the knot by the name Ashleys Bend in his 1947 book The Art of Knotting & Splicing just a few years after the publication of Ashleys book, later authors have continued to use this name. In the 1930s, Ashley performed security tests on a number of bends for the Collins, the manufacturer wanted a bend that would not slip when tied in mohair, a stiff slippery material. The jerk testing Ashley performed placed his bend, #1452, equal to the knot in exhibiting no slippage at all. All other bends he tested slipped to some extent, and most failed completely in less than 100 loading cycles, most references fail to distinguish the distinct ways in which the two ends of the knot can be dressed. List of bend knots List of knots

33.
Butterfly bend
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The butterfly bend is a knot used to join the ends of two ropes together. It is the bend form of the butterfly loop, in that it is the butterfly loop with the loop cut. The observation that the loop is secure enough to isolate a worn or damaged section of rope within the loop indicated that the bend form of the knot would be similarly secure. When Phil D. Smith made the first known presentation of the Hunters bend in 1953, while the bend form had been known to mountaineers, nautical rigger Brion Toss brought the knot to a wider audience when he published it in 1975. Unaware of the publication, Toss called the butterfly bend the strait bend after the Strait of Juan de Fuca. The butterfly bend can be tied using a subset of the used for tying the loop form by holding the two rope ends together and treating them as if they were a single bight. However, subtle positioning errors during the above shown tying method can result in a similar looking, a study of 8 different bends showed that the butterfly bend was the strongest and consistent across different ropes. The study, however, recommended the Double fishermans knot because the bend was almost impossible to untie after a significant load of about 1,000 lbs. was applied. Knot List of bend knots List of knots

34.
Basket weave knot
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The basket weave knots are a family of bend and lanyard knots with a regular pattern of over–one, under–one. All of these knots are rectangular and lie in a plane and they are named after plait-woven baskets, which have a similar appearance. A basket weave knot is made up of two sets of lines drawn inside a rectangle such that the lines meet at the edges of the rectangle. For a true basket weave knot that can be tied with two strands, the number of intersections in each direction cannot have a common divisor, within this constraint, there is no theoretical upper limit to the size of a basket weave knot. Thus, a knot that has two intersections in one direction can be lengthened with any odd number in the perpendicular direction, if the dimension n in the smaller direction is odd, it is always possible to construct a knot with n +2 intersections in the other dimension. However, large basket weave knots have a tendency to twist, a basket weave knot can be tied from a single strand by first forming a bight in the middle of the line. The ends near the bight become the standing ends and this method will keep the knot in one plane only for knots in which the standing ends enter the same side, these knots are called bosuns knots because they can be tied in a lanyard. For knots in which the standing ends enter from different sides of the rectangle, any basket weave knot that can be tied from two strands can be drawn as an endless knot by connecting the standing ends together and the working ends together. An example of this can be seen in the carrick mat, if a basket weave knot is tied with a flat line such as ribbon instead of a round line such as rope or cord, the method of turning the line at the edges affects the final appearance. Deflecting the line form a series of bights or scallops along the edge. Therefore, any of these knots could be used for a lanyard, in the carrick bend, which is otherwise similar to the double coin knot, the standing ends enter opposite long sides

35.
Beer knot
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A beer knot is a bend used to join tubular webbing. Its most common application is in constructing slings used in rock climbing, compared with the water knot, it has the advantages of a higher strength, smaller profile, and a cleaner appearance due to the lack of free-hanging tails. However, the knot can be more difficult to tie than the water knot. Testing by PMI in 1995 showed that the beer knot preserves about 80% of the strength of the webbing, the beer knot was introduced to the National Speleological Society in the 1980s by Peter Ludwig, from Austria. List of bend knots List of knots Video Instruction for Tying a Beer Knot

36.
Blood knot
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A blood knot is most usefully employed for joining sections of monofilament nylon line while maintaining a high portion of the lines inherent strength. Other knots used for this purpose can cause a loss of strength. In fly fishing, this serves to build a leader of gradually decreasing diameter with the fly line attached at the large diameter end. The principal drawback to the knot is the dexterity required to tie it. It is also likely to jam, which is not a concern in fishing line, which is no loss to cut. In tying the knot, the two lines to be joined are overlapped for 6–8 cm with the short ends of the two lines in opposite directions. The short end of one line is then wrapped 4–6 times around the second line and the remaining portion of the first short end brought back and passed between the lines at the beginning of the wraps. The short end of the line is then wrapped 4–6 times around the first line. The above method has been called by Stanle Barnes outcoil, and is contrasted with the method that resembles the finished knot from the start, incoil. In fishing line, and in other material if not deliberately set snug and maybe re-set after some initial tensioning, the lines are moistened and the wraps tightened by pulling on the long ends of the line. This causes the wraps to tighten and compress, creating two short sections of barrel, which much like a hangmans knot, that slide together. The short ends of the line are then trimmed close to the wraps, or one of the ends may be left intact to be used for a fly or lure. Half blood knot List of bend knots List of knots Video instructions for tying a Blood Knot Grog

37.
Double fisherman's knot
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The double fishermans knot or grapevine knot is a bend, or a knot used to join two lengths of rope. This knot and the fishermans knot are the variations used most often in climbing, arboriculture. The knot is formed by tying an overhand knot, in its strangle knot form. A primary use of this knot is to form high strength slings of cord for connecting pieces of a protection system. This knot, along with the basic fishermans knot can be used to join the ends of a necklace cord, the two strangle knots are left separated, and in this way the length of the necklace can be adjusted without breaking or untying the strand. A study of 8 different bends using climbing ropes concluded The Double Fishermans Knot seemed to be the best joining knot, the Butterfly bend held under slightly more weight, but almost impossible to untie after a significant load of about 1,000 lbs. was applied. List of bend knots List of knots Double Fishermans Knot Video

38.
Fisherman's knot
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The fishermans knot is a bend with a symmetrical structure consisting of two overhand knots, each tied around the standing part of the other. Other names for the fishermans knot include, anglers knot, English knot, halibut knot and it is compact, jamming when tightened and the working ends can be cropped very close to the knot. It can also be tied with cold, wet hands. Though these properties are suited to fishing, there are other knots which may provide superior performance. In knitting, the knot is used to join two strands of yarn, in this context, it is commonly known as the magic knot. True lovers knot List of bend knots List of knots

39.
Flemish bend
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The Flemish bend, also known as a figure eight bend, a double figure eight bend, and a rewoven figure eight is a knot for joining two ropes of roughly similar size. A loose figure-eight knot is tied in the end of one rope, the second rope is now threaded backwards parallel to the first rope. When properly dressed, the two strands do not cross each other, although fairly secure, it is susceptible to jamming. If tied, dressed and stressed properly it does not need stopper or safety knots, List of bend knots List of knots Flemish, or double figure eight, bend animated video by Marinews Grog

40.
Grief knot
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A grief knot is a knot which combines the features of a granny knot and a thief knot, producing a result which is not generally useful for working purposes. The word grief here is a portmanteau of granny and thief, the grief knot resembles the granny knot, but tied so that the working ends come out diagonally from each other, whereas a granny knots ends both come out on the same side. It unravels rather elegantly, as tension is applied, the ropes rotate like little cogs, to tie the grief knot, tie a single overhand knot, as if starting a reef knot. Then thrust the two ends together down through the center of the just-tied overhand knot. Twist the free ends to form half hitches to lock, twist the other way for the granny knot-like configuration that rolls apart when the parts are pulled. In short, if the standing parts nip/cross their own ends, the starkly differing behavior of the knot, depending on how it is arranged, has been exploited as the basis of a parlor trick. When pulling on the ends the knot starts slipping and the working ends become crossed. By twisting the ends so that they uncross and then recross in reverse. The twisting motion has been paralleled to the turning of a key, because the grief knot is known to slip apart with astonishing ease, it is considered one of the most insecure of knots. This is because the shape helps to prevent the knot from accidentally unlocking. When used in this manner, the knot is known as a grass bend, List of binding knots List of knots The Reef Knot Family

41.
Harness bend
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The harness knot is a general purpose bend knot used to join two ropes together. The knot can be tied under tension and will not capsize, the harness knot is essentially one half hitch and one crossing hitch each made by one of the two joined ropes, around the other ropes body. The ends get caught in between the two ropes and these two hitches, at the eye in the middle of the knot. There are two variants to this bend, a double harness bend with ends pointing in opposite directions. The starting side of one of the hitches has to be different, in order to have the approach the elliptical eye in the middle. The double harness bend with parallel ends is an unfinished Reefer bend, The ends need to go through the half hitch. The double harness bend with parallel ends is an unfinished Blood knot, the double harness bend is an unfinished Fishermans knot, the end needs to go through its own half hitch to form a overhand knot. All these knots are more secure than the harness knot but they are not as easy to untie, harness bend is useful when one needs to tighten the slack in a binding loop before locking the knot in the tight position. The name probably comes from the use in fixing the saddle on the back of the horse, tightening as soon as the horse that has learned to inhale at first move, exhales

42.
Heaving line bend
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The heaving line bend is a knot for securely joining two ropes of different diameter or rigidity. It is often used to affix playing strings to the thick silk eyes of a knot in some stringed instruments. In nautical use, the heaving line bend is used to connect a lighter messenger line to a hawser when mooring ships and it is knot number 1463 in The Ashley Book of Knots, and appeared in the 1916 Swedish knot manual Om Knutar. This avoid jamming when the line is pulled to carry the thick end over a distance where hands can not reach. Make a bight in the larger line, pass the lighter line around the standing part of the bight. Cross between the larger and the line on the back side. Finish by tucking the end between its turn around the part of the bight and that leg, pull tight. List of knots List of bend knots