Category: Knots

In mathematics, the braid group on n strands (denoted ${\displaystyle B_{n}}$), also known as the Artin braid group,^[1] is the group whose elements are equivalence classes of n-braids (e.g. under ambient isotopy), and whose group operation is composition of braids (see § Introduction). Example applications of braid groups include knot theory, where any knot may be represented as the closure of certain braids (a result known as Alexander’s theorem); in mathematical physics where Artin’s canonical presentation of the braid group corresponds to the Yang–Baxter equation (see § Basic properties); and in monodromy invariants of algebraic geometry.^[2]

In this introduction let n = 4; the generalization to other values of n will be straightforward. Consider two sets of four items lying on a table, with the items in each set being arranged in a vertical line, and such that one set sits next to the other. (In the illustrations below, these are the black dots.) Using four strands, each item of the first set is connected with an item of the second set so that a one-to-one correspondence results. Such a connection is called a braid. Often some strands will have to pass over or under others, and this is crucial: the following two connections are different braids:

is different from

On the other hand, two such connections which can be made to look the same by “pulling the strands” are considered the same braid:

is the same as

All strands are required to move from left to right; knots like the following are not considered braids:

is not a braid

Any two braids can be composed by drawing the first next to the second, identifying the four items in the middle, and connecting corresponding strands:

composed with

yields

Another example:

composed with

yields

The composition of the braids σ and τ is written as στ.

The set of all braids on four strands is denoted by ${\displaystyle B_{4}}$. The above composition of braids is indeed a group operation. The identity element is the braid consisting of four parallel horizontal strands, and the inverse of a braid consists of that braid which “undoes” whatever the first braid did, which is obtained by flipping a diagram such as the ones above across a vertical line going through its centre. (The first two example braids above are inverses of each other.)

Braid theory has recently been applied to fluid mechanics, specifically to the field of chaotic mixing in fluid flows. The braiding of (2 + 1)-dimensional space-time trajectories formed by motion of physical rods, periodic orbits or “ghost rods”, and almost-invariant sets has been used to estimate the topological entropy of several engineered and naturally occurring fluid systems, via the use of Nielsen–Thurston classification.^[3][4][5]

Another field of intense investigation involving braid groups and related topological concepts in the context of quantum physics is in the theory and (conjectured) experimental implementation of the proposed particles anyons. These have been proposed as the basis for error-corrected quantum computing and so their abstract study is currently of fundamental importance in quantum information.^[6]

To put the above informal discussion of braid groups on firm ground, one needs to use the homotopy concept of algebraic topology, defining braid groups as fundamental groups of a configuration space. Alternatively, one can define the braid group purely algebraically via the braid relations, keeping the pictures in mind only to guide the intuition.

To explain how to reduce a braid group in the sense of Artin to a fundamental group, we consider a connected manifold ${\displaystyle X}$ of dimension at least 2. The symmetric product of ${\displaystyle n}$ copies of ${\displaystyle X}$ means the quotient of ${\displaystyle X^{n}}$ (the ${\displaystyle n}$-fold Cartesian product of ${\displaystyle X}$) by the permutation action of the symmetric group on ${\displaystyle n}$ strands operating on the indices of coordinates. That is, an ordered ${\displaystyle n}$-tuple is in the same orbit as any other that is a re-ordered version of it.

A path in the ${\displaystyle n}$-fold symmetric product is the abstract way of discussing ${\displaystyle n}$ points of ${\displaystyle X}$, considered as an unordered ${\displaystyle n}$-tuple, independently tracing out ${\displaystyle n}$ strings. Since we must require that the strings never pass through each other, it is necessary that we pass to the subspace ${\displaystyle Y}$ of the symmetric product, of orbits of ${\displaystyle n}$-tuples of distinct points. That is, we remove all the subspaces of ${\displaystyle X^{n}}$ defined by conditions ${\displaystyle x_{i}=x_{j}}$ for all ${\displaystyle 1\leq i<j\leq n}$. This is invariant under the symmetric group, and ${\displaystyle Y}$ is the quotient by the symmetric group of the non-excluded ${\displaystyle n}$-tuples. Under the dimension condition ${\displaystyle Y}$ will be connected.

With this definition, then, we can call the braid group of ${\displaystyle X}$ with ${\displaystyle n}$ strings the fundamental group of ${\displaystyle Y}$ (for any choice of base point – this is well-defined up to isomorphism). The case where ${\displaystyle X}$ is the Euclidean plane is the original one of Artin. In some cases it can be shown that the higher homotopy groups of ${\displaystyle Y}$ are trivial.

Braid groups were introduced explicitly by Emil Artin in 1925, although (as Wilhelm Magnus pointed out in 1974^[11]) they were already implicit in Adolf Hurwitz’s work on monodromy from 1891.

Braid groups may be described by explicit presentations, as was shown by Artin in 1947.^[12] Braid groups are also understood by a deeper mathematical interpretation: as the fundamental group of certain configuration spaces.^[12]

As Magnus says, Hurwitz gave the interpretation of a braid group as the fundamental group of a configuration space (cf. braid theory), an interpretation that was lost from view until it was rediscovered by Ralph Fox and Lee Neuwirth in 1962.^[13]

Joan Birman’s book Braids, Links, and Mapping Class Groups (1974)^[14] was the first book devoted to braid groups.^[15]

In analogy with the action of the symmetric group by permutations, in various mathematical settings there exists a natural action of the braid group on n-tuples of objects or on the n-folded tensor product that involves some “twists”. Consider an arbitrary group G and let X be the set of all n-tuples of elements of G whose product is the identity element of G. Then B_n acts on X in the following fashion:

${\displaystyle \sigma _{i}\left(x_{1},\ldots ,x_{i-1},x_{i},x_{i+1},\ldots ,x_{n}\right)=\left(x_{1},\ldots ,x_{i-1},x_{i+1},x_{i+1}^{-1}x_{i}x_{i+1},x_{i+2},\ldots ,x_{n}\right).}$

Thus the elements x_i and x_i+1 exchange places and, in addition, x_i is twisted by the inner automorphism corresponding to x_i+1 – this ensures that the product of the components of x remains the identity element. It may be checked that the braid group relations are satisfied and this formula indeed defines a group action of B_n on X. As another example, a braided monoidal category is a monoidal category with a braid group action. Such structures play an important role in modern mathematical physics and lead to quantum knot invariants.

There are many ways to generalize this notion to an infinite number of strands. The simplest way is to take the direct limit of braid groups, where the attaching maps ${\displaystyle f\colon B_{n}\to B_{n+1}}$ send the ${\displaystyle n-1}$ generators of ${\displaystyle B_{n}}$ to the first ${\displaystyle n-1}$ generators of ${\displaystyle B_{n+1}}$ (i.e., by attaching a trivial strand). This group, however, admits no metrizable topology while remaining continuous.

Paul Fabel has shown that there are two topologies that can be imposed on the resulting group each of whose completion yields a different group.^[20] The first is a very tame group and is isomorphic to the mapping class group of the infinitely punctured disk—a discrete set of punctures limiting to the boundary of the disk.

The second group can be thought of the same as with finite braid groups. Place a strand at each of the points ${\displaystyle (0,1/n)}$ and the set of all braids—where a braid is defined to be a collection of paths from the points ${\displaystyle (0,1/n,0)}$ to the points ${\displaystyle (0,1/n,1)}$ so that the function yields a permutation on endpoints—is isomorphic to this wilder group. An interesting fact is that the pure braid group in this group is isomorphic to both the inverse limit of finite pure braid groups ${\displaystyle P_{n}}$ and to the fundamental group of the Hilbert cube minus the set

${\displaystyle \{(x_{i})_{i\in \mathbb {N} }\mid x_{i}=x_{j}{\text{ for some }}i\neq j\}.}$

The cohomology of a group ${\displaystyle G}$ is defined as the cohomology of the corresponding Eilenberg–MacLane classifying space, ${\displaystyle K(G,1)}$, which is a CW complex uniquely determined by ${\displaystyle G}$ up to homotopy. A classifying space for the braid group ${\displaystyle B_{n}}$ is the n^th unordered configuration space of ${\displaystyle \mathbb {R} ^{2}}$, that is, the space of all sets of ${\displaystyle n}$ distinct unordered points in the plane:^[21]

${\displaystyle \operatorname {UConf} _{n}(\mathbb {R} ^{2})=\{\{u_{1},…,u_{n}\}:u_{i}\in \mathbb {R} ^{2},u_{i}\neq u_{j}{\text{ for }}i\neq j\}}$.

So by definition ${\displaystyle H^{*}(B_{n})=H^{*}(K(B_{n},1))=H^{*}(\operatorname {UConf} _{n}(\mathbb {R} ^{2})).}$

The calculations for coefficients in ${\displaystyle \mathbb {Z} /2\mathbb {Z} }$ can be found in Fuks (1970).^[22]

Similarly, a classifying space for the pure braid group ${\displaystyle P_{n}}$ is ${\displaystyle \operatorname {Conf} _{n}(\mathbb {R} ^{2})}$, the n^th ordered configuration space of ${\displaystyle \mathbb {R} ^{2}}$. In 1968 Vladimir Arnold showed that the integral cohomology of the pure braid group ${\displaystyle P_{n}}$ is the quotient of the exterior algebra generated by the collection of degree-one classes ${\displaystyle \omega _{ij}\;\;1\leq i<j\leq n}$, subject to the relations^[23]

${\displaystyle \omega _{k,\ell }\omega _{\ell ,m}+\omega _{\ell ,m}\omega _{m,k}+\omega _{m,k}\omega _{k,\ell }=0.}$

• Affine symmetric group, a braid group on an annulus
• Artin–Tits group
• Braided Hopf algebra
• Braided monoidal category
• Braided vector space
• Knot theory
• Non-commutative cryptography
• Spherical braid group

• “Braid group”. PlanetMath.
• CRAG: CRyptography and Groups computation library from the Stevens University’s Algebraic Cryptography Center
• Macauley, M. Lecture 1.3: Groups in science, art, and mathematics. Visual Group Theory. Clemson University.
• Bigelow, Stephen. “Exploration of B5 Java applet”. Archived from the original on 4 June 2013. Retrieved 1 November 2007.
• Lipmaa, Helger, Cryptography and Braid Groups page, archived from the original on 3 August 2009
• Dalvit, Ester (2015). Braids – the movie.
• Scherich, Nancy. Representations of the Braid Groups. Dance Your PhD. expanded further in Behind the Math of “Dance Your PhD,” Part 1: The Braid Groups.

Rigid double splayed loop in the bight
Names	Rigid double splayed loop in the bight, double splayed loop
Category	Loop
Related	Alpine butterfly knot
ABoK	#1100

The rigid double splayed loop in the bight is a knot that contains two parallel loops. Clifford Ashley wrote that it is “one of the firmest of the Double Loops since the two loops do not directly communicate with each other”.^[1] (In actuality, it can be argued that the two loops do directly communicate as the two center portions of each loop simply pass down through the head knot and pass around the running ends; not significantly different, in that regard, from the Spanish Bowline). It is a variation of the alpine butterfly knot.

This knot can be tied in the bight as C. Ashley explains, but it can also be tied in the end around objects in a simple way. This makes it suitable for improvising a harness or for slinging a ladder for a staging. This knot is simply a pair of intertwined left-handed bowlines or cowboy bowlines (ABOK #1034½) that share an element in common. To tie it:

• 1- Tie a left-hand bowline around the first object and pass the end of the rope through the second. Do not tighten the knot.
• 2- Pass the end of the rope through the hole marked in green in the first photo, from behind.
• 3- Complete a half hitch by passing the end through the second loop.
• 4- Finish by threading the end of the rope through the top loop parallel to the standing part. Now it is time to adjust the size of both loops as desired and then tighten the knot.
• 5-6- Optional: add a half hitch or finish the knot with a regular bowline, similar to ABOK #1075.

Replacing the cowboy bowline with a normal bowline (ABOK #1010) in the above procedure will give you a variant of the knot that only differs from the original in one additional crossing.

• Double Loop Knots: A Rigid Double Splayed Loop in the Bight

In the mathematical area of knot theory, a ribbon knot is a knot that bounds a self-intersecting disk with only ribbon singularities. Intuitively, this kind of singularity can be formed by cutting a slit in the disk and passing another part of the disk through the slit. More precisely, this type of singularity is a closed arc consisting of intersection points of the disk with itself, such that the preimage of this arc consists of two arcs in the disc, one completely in the interior of the disk and the other having its two endpoints on the disk boundary.

A slice disc M is a smoothly embedded ${\displaystyle D^{2}}$ in ${\displaystyle D^{4}}$ with ${\displaystyle M\cap \partial D^{4}=\partial M\subset S^{3}}$. Consider the function ${\displaystyle f\colon D^{4}\to \mathbb {R} }$ given by ${\displaystyle f(x,y,z,w)=x^{2}+y^{2}+z^{2}+w^{2}}$. By a small isotopy of M one can ensure that f restricts to a Morse function on M. One says ${\displaystyle \partial M\subset \partial D^{4}=S^{3}}$ is a ribbon knot if ${\displaystyle f_{|M}\colon M\to \mathbb {R} }$ has no interior local maxima.

Every ribbon knot is known to be a slice knot. A famous open problem, posed by Ralph Fox and known as the slice-ribbon conjecture, asks if the converse is true: is every (smoothly) slice knot ribbon?

Lisca (2007) showed that the conjecture is true for knots of bridge number two. Greene & Jabuka (2011) showed it to be true for three-stranded pretzel knots with odd parameters. However, Gompf, Scharlemann & Thompson (2010) suggested that the conjecture might not be true, and provided a family of knots that could be counterexamples to it. The conjecture was further strengthened when a famous potential counter-example, the (2, 1) cable of the figure-eight knot, was shown to be not slice and thereby not a counterexample.^[1][2]

• Fox, R. H. (1962), “Some problems in knot theory”, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961), Englewood Cliffs, New Jersey: Prentice-Hall, pp. 168–176, MR 0140100. Reprinted by Dover Books, 2010.
• Gompf, Robert E.; Scharlemann, Martin; Thompson, Abigail (2010), “Fibered knots and potential counterexamples to the property 2R and slice-ribbon conjectures”, Geometry & Topology, 14 (4): 2305–2347, arXiv:1103.1601, doi:10.2140/gt.2010.14.2305, MR 2740649, S2CID 58915479.
• Greene, Joshua; Jabuka, Stanislav (2011), “The slice-ribbon conjecture for 3-stranded pretzel knots”, American Journal of Mathematics, 133 (3): 555–580, arXiv:0706.3398, doi:10.1353/ajm.2011.0022, MR 2808326, S2CID 10279100.
• Kauffman, Louis H. (1987), On Knots, Annals of Mathematics Studies, vol. 115, Princeton, New Jersey: Princeton University Press, ISBN 0-691-08434-3, MR 0907872.
• Lisca, Paolo (2007), “Lens spaces, rational balls and the ribbon conjecture”, Geometry & Topology, 11: 429–472, arXiv:math/0701610, doi:10.2140/gt.2007.11.429, MR 2302495, S2CID 15238217.

• Sloman, Leila (18 May 2022). “How Complex Is a Knot? New Proof Reveals Ranking System That Works”. Quanta Magazine.

Braids (also referred to as plaits) are a hairstyle formed by interlacing three or more strands of hair.^[1] Braiding has been used to style and ornament human and animal hair for thousands of years world-wide^[2] in various cultures around the world. Braided hairstyles are used in many cultures as social signifiers for traits such as marital status, gender, social class, and age.

The simplest and most common version is a flat, solid, three-stranded structure. More complex patterns can be constructed from an arbitrary number of strands to create a wider range of structures. The structure is usually long and narrow with each component strand functionally equivalent in zigzagging forward through the overlapping mass of the others. Incorporating more hair as the braid progresses, either from the head or from separate wefts of hair, can create other styles such as knotless braids and French braids. Small items like beads and shells can also be incorporated into the braid.

The earliest known depiction of braiding may be the Venus of Willendorf, a female figurine unearthed in Austria which is estimated to have been made between about 28,000 and 25,000 BCE.^[3] It has been disputed whether the horizontal spiral ridges on the statue’s head depict braided hair or some sort of a woven basket.^[4]
The Venus of Brassempouy in France is estimated to be about 25,000 years old and ostensibly shows a braided hairstyle.^[5]

Various braided hairstyles have historically been common in cultures throughout the world. Braids of varying styles have been extant in Western Asia, the Indian sub-continent and China since the Bronze Age. In Western Asia braiding of hair and beards was commonplace throughout the Bronze Age and Iron Age in Mesopotamia, Levant, Iran and Anatolia, as evidenced in the art of Sumer, Akkad Assyria, Babylonia, Elam, Ebla, and the Canaanites, Hittites, Hurrians, Israelites, Phoenicians, Persians, Armenians among others.

Bog bodies in Northern Europe have been found wearing braided hairstyles from the Northern European Iron Age, and later still braided styles were found among the Celts, Iberians, Germanic peoples, Slavs and Vikings in northern, western, Eastern and southwestern Europe.^[6][7] The people of Ancient Egypt, Greece, and Rome all wore braids.^[8][9]

In some regions, a braid was a means of communication. At a glance, one individual could distinguish a wealth of information about another, whether they were married, mourning, or of age for courtship, simply by observing their hairstyle. Braids were a means of social stratification. Certain hairstyles were distinctive to particular ethnicities or nations. Other styles informed others of an individual’s status in society. Braid patterns or hairstyles could indicate a person’s community, age, marital status, wealth, power, social position, and religion.^[10]

Braiding is traditionally a social art. Because of the time it takes to braid hair, people have often taken time to socialize while braiding and having their hair braided. It begins with the elders making simple knots and braids for younger children. Older children watch and learn from them, start practicing on younger children, and eventually learn the traditional designs. This carries on a tradition of bonding between elders and the new generation.^[11]

There are a number of different types of braided hairstyles, including French braids, corn rows, and box braiding.^[12] Braided hairstyles may also be used in combination with or as an alternative to simpler bindings, such as ponytails or pigtails. Braiding may also be used to add ornamentation, such as beads or hair extensions, as in crochet braiding.

The cutting of braids is a powerful symbol in many cultures. The forced cutting of braids has also been used as a form of punishment.

In Qing Dynasty China, cutting the braided queue was a symbol of resistance against Manchu rule.^[40] Under the Republic of China, men were required to cut their braids.

In many Native American tribes, one’s braided hair is cut when mourning the loss of a loved one. Under the Canadian Indian residential school system, Indigenous children’s braids were often cut to erase their cultural origins.^[45] Cutting braids has been used as a form of protest against the appropriation of Native and First Nations land.^[46]

In 2017, a spate of braid-cutting attacks occurred in Northern India. No explanation has emerged for the attacks.

Braiding is also used to prepare horses’ manes and tails for showing, often for show jumping, polo and dressage.^[56] Braiding horse hair can sometimes be beneficial since it can protect manes and tails from damage.^[57]

• African hair threading
• Box braids
• Braid
• Koroba
• List of hairstyles
• Shuku

Reidemeister moves

Type I	Type II	Type III

Modified Reidemeister move

Type I’

In the mathematical area of knot theory, a Reidemeister move is any of three local moves on a link diagram. Kurt Reidemeister (1927) and, independently, James Waddell Alexander and Garland Baird Briggs (1926), demonstrated that two knot diagrams belonging to the same knot, up to planar isotopy, can be related by a sequence of the three Reidemeister moves.^[1][2]

Each move operates on a small region of the diagram and is one of three types:^[3]

No other part of the diagram is involved in the picture of a move, and a planar isotopy may distort the picture. The numbering for the types of moves corresponds to how many strands are involved, e.g. a type II move operates on two strands of the diagram.

One important context in which the Reidemeister moves appear is in defining knot invariants.^[4] By demonstrating a property of a knot diagram which is not changed when we apply any of the Reidemeister moves, an invariant is defined. Many important invariants can be defined in this way, including the Jones polynomial.

The type I move is the only move that affects the writhe of the diagram. The type III move is the only one which does not change the crossing number of the diagram.^[5]

In applications such as the Kirby calculus, in which the desired equivalence class of knot diagrams is not a knot but a framed link, one must replace the type I move with a “modified type I” (type I’) move composed of two type I moves of opposite sense. The type I’ move affects neither the framing of the link nor the writhe of the overall knot diagram.^[6]

Trace (1983) showed that two knot diagrams for the same knot are related by using only type II and III moves if and only if they have the same writhe and winding number.^[7] Furthermore, combined work of Östlund (2001), Manturov (2004), and Hagge (2006) shows that for every knot type there are a pair of knot diagrams so that every sequence of Reidemeister moves taking one to the other must use all three types of moves.^[8] Alexander Coward demonstrated that for link diagrams representing equivalent links, there is a sequence of moves ordered by type: first type I moves, then type II moves, type III, and then type II. The moves before the type III moves increase crossing number while those after decrease crossing number.

Hayashi (2005) proved there is also an upper bound, depending on crossing number, on the number of Reidemeister moves required to split a link.^[9]

Coward & Lackenby (2014) proved the existence of an exponential tower upper bound (depending on crossing number) on the number of Reidemeister moves required to pass between two diagrams of the same link.^[10] In detail, let ${\displaystyle n}$ be the sum of the crossing numbers of the two diagrams, then the upper bound is ${\displaystyle 2^{2^{2^{.^{.^{n}}}}}}$ where the height of the tower of ${\displaystyle 2}$s (with a single ${\displaystyle n}$ at the top) is ${\displaystyle 10^{1,000,000n}}$.

Lackenby (2015) proved the existence of a polynomial upper bound (depending on crossing number) on the number of Reidemeister moves required to change a diagram of the unknot to the standard unknot. In detail, for any such diagram with ${\displaystyle c}$ crossings, the upper bound is ${\displaystyle (236c)^{11}}$.^[11]

• Alexander, James W.; Briggs, Garland B. (1926), “On types of knotted curves”, Annals of Mathematics, 28 (1/4): 562–586, doi:10.2307/1968399, JSTOR 1968399, MR 1502807
• Coward, Alexander; Lackenby, Marc (2014), “An upper bound on Reidemeister moves”, American Journal of Mathematics, 136 (4): 1023–1066, arXiv:1104.1882, doi:10.1353/ajm.2014.0027, MR 3245186, S2CID 55882290
• Galatolo, Stefano (1999), “On a problem in effective knot theory”, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 9 (4): 299–306, MR 1722788
• Hagge, Tobias (2006), “Every Reidemeister move is needed for each knot type”, Proc. Amer. Math. Soc., 134 (1): 295–301, doi:10.1090/S0002-9939-05-07935-9, MR 2170571
• Hass, Joel; Lagarias, Jeffrey C. (2001), “The number of Reidemeister moves needed for unknotting”, Journal of the American Mathematical Society, 14 (2): 399–428, arXiv:math/9807012, doi:10.1090/S0894-0347-01-00358-7, MR 1815217, S2CID 15654705
• Hayashi, Chuichiro (2005), “The number of Reidemeister moves for splitting a link”, Mathematische Annalen, 332 (2): 239–252, doi:10.1007/s00208-004-0599-x, MR 2178061, S2CID 119728321
• Lackenby, Marc (2015), “A polynomial upper bound on Reidemeister moves”, Annals of Mathematics, Second Series, 182 (2): 491–564, arXiv:1302.0180, doi:10.4007/annals.2015.182.2.3, MR 3418524, S2CID 119662237
• Manturov, Vassily Olegovich (2004), Knot theory, Boca Raton, FL: Chapman & Hall/CRC, doi:10.1201/9780203402849, ISBN 0-415-31001-6, MR 2068425
• Östlund, Olof-Petter (2001), “Invariants of knot diagrams and relations among Reidemeister moves”, J. Knot Theory Ramifications, 10 (8): 1215–1227, arXiv:math/0005108, doi:10.1142/S0218216501001402, MR 1871226, S2CID 119177881
• Reidemeister, Kurt (1927), “Elementare Begründung der Knotentheorie”, Abh. Math. Sem. Univ. Hamburg, 5 (1): 24–32, doi:10.1007/BF02952507, MR 3069462, S2CID 120149796
• Trace, Bruce (1983), “On the Reidemeister moves of a classical knot”, Proceedings of the American Mathematical Society, 89 (4): 722–724, doi:10.2307/2044613, JSTOR 2044613, MR 0719004

A braid (also referred to as a plait; /plæt/) is a complex structure or pattern formed by interlacing three or more strands of flexible material such as textile yarns, wire, or hair.^[1]
The simplest and most common version is a flat, solid, three-stranded structure. More complex patterns can be constructed from an arbitrary number of strands to create a wider range of structures (such as a fishtail braid, a five-stranded braid, rope braid, a French braid and a waterfall braid). The structure is usually long and narrow with each component strand functionally equivalent in zigzagging forward through the overlapping mass of the others. It can be compared with the process of weaving, which usually involves two separate perpendicular groups of strands (warp and weft).

Historically, the materials used have depended on the French plants and animals available in the local area. During the Industrial Revolution, mechanized braiding equipment was invented to increase production. The braiding technique was used to make ropes^[2] with both natural and synthetic fibers as well as coaxial cables for radios using copper wire.^[3] In more recent times it has been used to create a covering for fuel pipes in jet aircraft and ships (first using glass fibre, then stainless steel and Kevlar). Hoses for domestic plumbing are often covered with stainless steel braid.

The oldest known reproduction of hair braiding may go back about 30,000 years: the Venus of Willendorf, a female figurine estimated to have been made between about 28,000 and 25,000 BC in modern-day Austria.^[4] The Venus of Brassempouy from the southwest of France is estimated to be about 25,000 years old and shows a braided hairstyle.

Braids originated in Africa, with the earliest evidence dating back to 3500 BCE, particularly among the Himba people of Namibia. This ancient practice, which is over 5,000 years old, held deep cultural, social, and religious significance, representing a person’s tribe, age, marital status, and wealth. Like how different versions of Cinderella are traceable to nearly every culture, braids, too, are polygenetic. One early example of hair braiding takes place in 1279-1213 BCE as recorded in the story of Isis: “when some of the queen’s maidens came to the well, she greeted them kindly and began to braid their hair.”^[5]

During the Bronze Age and Iron Age, many peoples in the Near East, Asia Minor, Caucasus, East Mediterranean and North Africa are depicted in art with braided or plaited hair and beards. Similarly, the practice is recorded in Europe, Africa, India, China, Japan, Australasia and Central Asia.

Braiding is traditionally a social art. Because of the time it takes to braid hair, people have often taken time to socialize while braiding and having their hair braided. It begins with the elders making simple knots and braids for younger children. Older children watch and learn from them, start practicing on younger children, and eventually learn the traditional designs. This carries on a tradition of bonding between elders and the new generation.

Early braids had many uses, such as costume decoration, animal regalia (like camel girths), sword decoration, bowls and hats (from palm leaves), locks (such as those made in Japan to secure precious tea supplies through the use of elaborate knots), and weapons (e.g. slings).

Materials that are used in braids can vary depending on local materials. For instance, South Americans used the very fine fibers from the wool of alpaca and llama, while North American people made use of bison fibers. Throughout the world, vegetable fibers such as grass, nettle, and hemp have been used to create braids. In China, Korea, and Japan silk still remains the main material used. In the Americas, the braiding of leather is also common. Plaiting with kangaroo leather has been a widely practiced tradition in rural Australia since pioneering times. It is used in the production of fine leather belts, hatbands, bridles, dog leads, bullwhips, stockwhips, etc. Other leathers are used for the plaiting of heavier products suitable for everyday use.^[6]

For nomadic peoples, braiding was a practical means of producing useful and decorative textiles. In other areas, such as the Pacific islands (where leaves and grasses are braided), and for many hill tribes, braids are made using minimal equipment. It was only when braiding became a popular occupation in the home or school, as it is in China and Japan, and when the Industrial Revolution came about, that specific tools were developed to increase production and make it easier to produce more complicated patterns of braids.

Braids are also very good for making rope and decorative objects.^[7] Complex braids have been used to create hanging fibre artworks.

Gold braids and silver braids are components or trims of many kinds of formal dress, including military uniform (in epaulettes, aiguillettes, on headgear).

Ropes and cables

Braiding creates a composite rope that is thicker than the non-interlaced strands of yarns. Braided ropes are preferred by arborists, rock climbers, and in sport sailing because they do not twist under load, as does an ordinary twisted-strand rope. These ropes consist of one or more concentric tubular braided jackets surrounding either several small twisted fibre cords, or a single untwisted yarn of straight fibres, and are known as Kernmantle ropes.

In electrical and electronic cables, braid is a tubular sheath made of braided strands of metal placed around a central cable for shielding against electromagnetic interference. The braid is grounded while the central conductor(s) carries the signal. The braid may be used in addition to a foil jacket to increase shielding and durability. Litz wire uses braids of thin insulated wires to carry high frequency signals with much lower losses from skin effect or to minimise proximity effect in transformers. Flat braids made of many copper wires can also be used for flexible electrical connections between large components. The numerous smaller wires comprising the braid are much more resistant to breaking under repeated motion and vibration than is a cable of larger wires.

Similar braiding is used on pressurized hoses, such as in plumbing and hydraulic brake systems in automobiles. Braiding is also used for fibres for composite reinforcements.

A property of the basic braid is that removing one strand unlinks the other two, as they are not twisted around each other. Mathematically, a braid with that property is called a Brunnian braid.

Onion and garlic stalks are often braided for storage after they are partially dried.^[8][9][10]

Braids are often used figuratively to represent interweaving or combination, such as in, “He braided many different ideas into a new whole.”

In some river and stream systems, small streams join and redivide in many places. Such stream systems are said to be braided.^[11] These are often found in alluvial fans at the outlet of canyons. This is a result of heavy sediment deposition at high flows followed by re-erosion at low flows.

• 
  A coaxial cable with braided copper wire EMI shielding (B)
• 
  A gold braid on a police uniform
• 
  A close up of a red braided USB cable.

• African hair threading
• Braiding machines
• Braid theory
• Ceinture fléchée
• Kumihimo, Japanese braid
• List of hairstyles
• Lucet
• Ply-split braiding
• Sennit

• Media related to Braids at Wikimedia Commons
• The dictionary definition of braid at Wiktionary
• Minnesota Statute Sec.155A.28 https://www.revisor.mn.gov/statutes/cite/155A.28

In the mathematical field of knot theory, the bracket polynomial (also known as the Kauffman bracket) is a polynomial invariant of framed links. Although it is not an invariant of knots or links (as it is not invariant under type I Reidemeister moves), a suitably “normalized” version yields the famous knot invariant called the Jones polynomial. The bracket polynomial plays an important role in unifying the Jones polynomial with other quantum invariants. In particular, Kauffman’s interpretation of the Jones polynomial allows generalization to invariants of 3-manifolds.

The bracket polynomial was discovered by Louis Kauffman in 1987.

The bracket polynomial of any (unoriented) link diagram ${\displaystyle L}$, denoted ${\displaystyle \langle L\rangle }$, is a polynomial in the variable ${\displaystyle A}$, characterized by the three rules:

• ${\displaystyle \langle \bigcirc \rangle =1}$, where ${\displaystyle \bigcirc }$ is the standard diagram of the unknot
• 
• ${\displaystyle \langle \bigcirc \sqcup L\rangle =(-A^{2}-A^{-2})\langle L\rangle }$

The pictures in the second rule represent brackets of the link diagrams which differ inside a disc as shown but are identical outside. The third rule means that adding a circle disjoint from the rest of the diagram multiplies the bracket of the remaining diagram by ${\displaystyle -A^{2}-A^{-2}}$.

• Louis H. Kauffman, State models and the Jones polynomial. Topology 26 (1987), no. 3, 395–407. (introduces the bracket polynomial)

• Weisstein, Eric W. “Bracket Polynomial”. MathWorld.

In the mathematical subject of knot theory, regular isotopy is the equivalence relation of link diagrams that is generated by using the 2nd and 3rd Reidemeister moves only. The notion of regular isotopy was introduced by Louis Kauffman (Kauffman 1990). It can be thought of as an isotopy of a ribbon pressed flat against the plane which keeps the ribbon flat. For diagrams in the plane this is a finer equivalence relation than ambient isotopy of framed links, since the 2nd and 3rd Reidemeister moves preserve the winding number of the diagram (Kauffman 1990, pp. 450ff.). However, for diagrams in the sphere (considered as the plane plus infinity), the two notions are equivalent, due to the extra freedom of passing a strand through infinity.

• Ambient isotopy
• Knot polynomial

• Kauffman, Louis H. (1990). “An invariant of regular isotopy”. Transactions of the American Mathematical Society. 318 (2): 417–471. doi:10.1090/S0002-9947-1990-0958895-7. ISSN 0002-9947. S2CID 53521408.

Reever Knot
Category	Bend
Related	Vice Versa Bend, Simple Simon Over, Simple Simon Under, Double Harness Bend
Releasing	Non-jamming

The Reever Knot is a secure bend for joining two ropes. An important attribute of the knot is that each line going in and out of the knot is clamped at two points within the knot. For this reason it is considered secure and resistant to being shaken loose when subject to intermittent loads.^[1]

The Reever Knot is closely related to the Vice Versa Bend. They only differ in the selection of which lines are used as the standing and working ends of the knot.

Given the structure of the knot there are three possible combinations one can use for the standing and working ends of the knot. The standing parts can be A-A, A-B, or B-B.
The Reever knot results when the standing ends are selected as A-A.^[2] Selecting the standing ends as A-B results in the Vice Versa Bend.^[3]

• 
  Reever knot: with standing ends A-A
• 
  Vice Versa bend: with standing ends A-B

All forms of the knot are considered reliable and secure but it is suggested that the Reever Knot is the better version because the arrangement of standing and working ends in the Vice Versa Bend is not strictly symmetric.^[1]

The Reever Knot appears in an article by C E I Wright and J E Magowan in volume 40 of the Alpine Journal in 1928 as a knot that is recommended for joining two ropes.^[2]

The Vice Versa Bend appears in The Alternative Knot Book by Harry Asher (1989). In the introduction to his ‘New System of Knots’ he presents a sequence of three new knots, the Simple Simon Over, the Simple Simon Under, and the Vice Versa Bend.
The three knots form a developmental sequence that were inspired by aspects of the Sheet bend.^[3]

In his 1995 book, Symmetric Bends: How to Join Two Lengths of Cord, Miles presents a knot theoretic analysis of 60 symmetric bends. The Vice Versa Bend appears as number 19 in this sequence. Miles attributes the knot to Asher and describes it as a ‘pure lanyard bend’ in which “two ends of equal status emerge from the knot in each of two opposite directions”.^[4]

Budworth, a founding member of the International Guild of Knot Tyers, includes the Vice Versa Bend in his 2000 book The Book of Practical Knots.
He also attributes the knot to Asher.^[5]

The relationship between the Reever Knot and the Vice Versa Bend was first pointed out by Clements In his 2004 article “The Vice Versa Bend and the Reever Knot”.^[1] His analysis of the symmetry of the two forms of the knot led him to suggest that the Reever Knot, being completely symmetric, is the better version of the knot. He concludes that the Reever knot is a secure bend that is compact and streamlined in form, and that it deserves to be more widely known and used.

The knot provides a compact, streamlined and decorative way of joining two ropes. However its primary attribute is that it is resistant against working loose when subject to intermittent loads.^[1] The security of the knot arises from the fact that at step 3 in the tying sequence the knot is a Double Harness Bend with parallel ends (ABoK #1421). The additional step of passing the ends through the outer loops to complete the knot results in each line entering and exiting the knot being clamped at two points within the knot.

• List of knots

Reef knot
Names	Reef knot, Square knot, Hercules knot, Double knot, brotherhood knot
Category	Binding
Origin	Ancient
Related	Thief knot, Granny knot, Grief knot, Surgeon’s knot, Shoelace knot
Releasing	Jamming
Typical use	Joining two ends of a single line to bind around an object.
Caveat	Not secure as a bend unless secured by additional knots(ex: overhand). Spills easily if one of the free ends is pulled outward. Does not hold well if the two lines are not the same thickness.
ABoK	#74, #75, #460, #1204, #1402, #2096, #2573, #2574, #2577, #2580
Instructions

The reef knot, or square knot, is an ancient and simple binding knot used to secure a rope or line around an object. It is sometimes also referred to as a Hercules knot or Heracles knot. The knot is formed by tying a left-handed overhand knot between two ends, instead of around one end, and then a right-handed overhand knot via the same procedure, or vice versa. A common mnemonic for this procedure is “right over left; left over right”, which is often appended with the rhyming suffix “… makes a knot both tidy and tight”. Two consecutive overhands tied as described above of the same handedness will make a weak granny knot. The working ends of the reef knot must emerge both at the top or both at the bottom, otherwise a very weak thief knot results.

The reef knot or square knot consists of two half knots, one left and one right, one being tied on top of the other, and either being tied first…The reef knot is unique in that it may be tied and tightened with both ends. It is universally used for parcels, rolls and bundles. At sea it is always employed in reefing and furling sails and stopping clothes for drying. But under no circumstances should it ever be tied as a bend, for if tied with two ends of unequal size, or if one end is stiffer or smoother than the other, the knot is almost bound to spill. Except for its true purpose of binding it is a knot to be shunned.

The reef knot is not recommended for tying two ropes together, because of the potential instability of the knot when not stabilized; this has resulted in many accidental deaths (see Misuse as a bend).

The reef knot is at least 4,000 years old. The name “reef knot” dates from at least 1794^[2] and originates from its common use to reef sails,^[3][4] that is to tie part of the sail down to decrease its effective surface area in strong winds.
The name “square knot” is found in Dana’s 1841 maritime compendium A Seaman’s Friend, which also gives “reef knot” as an alternative name.^[5][6]

The name square knot is often used for the unslipped version of reef knot. Reef knot itself then is understood as the single slipped version, while the name shoelace knot is to indicate double slipped version.

Sometimes the name bowtie also may be used to indicate a double slipped version, but tying a bowtie is usually performed on flat material, and involves a slip knot of one end holding a bight of the other end i.e. not really a double slipped reef knot.

The name “Square knot” is also used for completely different other knots such as the mathematical concept of square knot, or friendship knot; this last one earns the name by being flat and drawing a square on one face (and a cross on the other face).

The reef knot is used to tie the two ends of a single rope together such that they will secure something, for example a bundle of objects, that is unlikely to move much.

The single slipped version is used by sailors for reefing and furling sails. To release the knot a sailor could collapse it with a pull of one hand on the slipped end. The sail’s weight would make the collapsed knot come apart. It is specifically this behavior which makes the knot unsafe for connecting two ropes together.^[7]

The reef knot is also one of the key knots of macrame textiles.^[8]

The knot lies flat when made with cloth and has been used for tying bandages for millennia. As a bandage knot it was known to the ancient Greeks as the Hercules knot (Herakleotikon hamma). and is still used extensively in medicine.
In his Natural History, Pliny relates the belief that wounds heal more quickly when bound with a Hercules knot.^[9][10]

It has also been used since ancient times to tie belts and sashes. A modern use in this manner includes tying the obi (or belt) of a martial arts keikogi.

With both ends slipped it becomes a good way to tie shoelaces, whilst the non-slipped version is useful for shoelaces that are excessively short. It is appropriate for tying plastic garbage or trash bags, as the knot forms a handle when tied in two twisted edges of the bag.

A surgeon’s variation, used where a third hand is unavailable, is made with two or three twists of the ropes on bottom, and sometimes on top, instead of just one.

The reef knot figures prominently in Scouting worldwide. It is included in the international membership badge and many scouting awards. In Pioneering (Scouting), it is commonly used as a binding knot to finish off specialized lashing (ropework) and whipping knots. However, it is an insecure knot, unstable when jiggled, and is not suitable for supporting weight.^[11][12][13]

• 
  Detail of Egyptian statue dating from 2350 BC depicting a reef knot securing a belt
• 
  Ancient Greek jewelry from Pontika (Ukraine), 300 BC, in the form of a reef knot
• 
  Weight for weighing gold dust – Knot – MHNT

The reef knot’s familiarity, ease of tying, and visually appealing symmetry conceal its weakness. The International Guild of Knot Tyers warns that this knot should never be used to bend two ropes together.^[14] However, modern instruction teaches that it is fine for noncritical applications,^[15] especially if stabilized. A proper bend knot, for instance a sheet bend or double fisherman’s knot, should be used instead. Knotting authority Clifford Ashley claimed that failures of misused reef knots have caused more deaths and injuries than failures of all other knots combined.^[16] Further, it is easily confused with the granny knot, which is a very poor knot.

An approximate physical analysis^[17] predicts that a reef knot will hold if ${\displaystyle 2\mu e^{\mu \pi }\geq 1}$, where μ is the relevant coefficient of friction. This inequality holds if ${\displaystyle \mu \gtrsim 0.24}$. Experiments show that the critical value of μ is actually somewhat lower.^[18]

• 
  Reef (square) knot
• 
  Thief knot
• 
  Granny knot
• 
  Grief knot
• 
  Surgeon’s knot

• Aberdeen knot (German Wikipedia)
• Shoelace knot
• Granny knot
• Thief knot
• Surgeon’s knot
• List of binding knots
• List of knots
• Square knot (emblem or insignia)
• Reefing

• Knot of Hercules
• Ancient symbolism of Hercules knot