Category: Knots

Square knot
Three-dimensional view
Common name	Reef knot
Crossing no.	6
Stick no.	8
A–B notation	${\displaystyle 3_{1}\#3_{1}^{*}}$
Other
alternating, composite, pretzel, slice, amphichiral, tricolorable

In knot theory, the square knot is a composite knot obtained by taking the connected sum of a trefoil knot with its reflection. It is closely related to the granny knot, which is also a connected sum of two trefoils. Because the trefoil knot is the simplest nontrivial knot, the square knot and the granny knot are the simplest of all composite knots.

The square knot is the mathematical version of the common reef knot.

The square knot can be constructed from two trefoil knots, one of which must be left-handed and the other right-handed. Each of the two knots is cut, and then the loose ends are joined together pairwise. The resulting connected sum is the square knot.

It is important that the original trefoil knots be mirror images of one another. If two identical trefoil knots are used instead, the result is a granny knot.

The square knot is amphichiral, meaning that it is indistinguishable from its own mirror image. The crossing number of a square knot is six, which is the smallest possible crossing number for a composite knot.

The Alexander polynomial of the square knot is

${\displaystyle \Delta (t)=(t-1+t^{-1})^{2},\,}$

which is simply the square of the Alexander polynomial of a trefoil knot. Similarly, the Alexander–Conway polynomial of a square knot is

${\displaystyle \nabla (z)=(z^{2}+1)^{2}.}$

These two polynomials are the same as those for the granny knot. However, the Jones polynomial for the square knot is

${\displaystyle V(q)=(q^{-1}+q^{-3}-q^{-4})(q+q^{3}-q^{4})=-q^{3}+q^{2}-q+3-q^{-1}+q^{-2}-q^{-3}.\,}$

This is the product of the Jones polynomials for the right-handed and left-handed trefoil knots, and is different from the Jones polynomial for a granny knot.

The knot group of the square knot is given by the presentation

${\displaystyle \langle x,y,z\mid xyx=yxy,xzx=zxz\rangle .\,}$^[1]

This is isomorphic to the knot group of the granny knot, and is the simplest example of two different knots with isomorphic knot groups.

Unlike the granny knot, the square knot is a ribbon knot, and it is therefore also a slice knot.

Girth hitch
Names	Girth hitch, Lark’s head, Lark’s foot, Ring hitch, Lanyard hitch, Bale Sling hitch, Baggage Tag Loop, Tag Knot, Deadeye hitch, Running eye, Strop Bend
Category	Hitch
Origin	Ancient
Related	Clove hitch, Cat’s paw, Bale sling hitch, Prusik
Releasing	Non-jamming
Typical use	Tying a rope to a ring or pole
Caveat	Can fail unless equal tension is applied to both of the standing parts of the rope.
ABoK	#5, #56, #59, #244, #310, #1184, #1673, #1694, #1698, #1700, #1802, #2163, #2164, #2168, #2175, #3317

The cow hitch, also called the lark’s head, is a hitch knot used to attach a rope to an object. The cow hitch comprises a pair of single hitches tied in opposing directions, as compared to the clove hitch in which the single hitches are tied in the same direction. It has several variations and is known under a variety of names. It can be tied either with the end of the rope or with a bight.

A simple and useful knotted structure, the cow hitch has been known since at least the first century when described by Greek physician Heraklas in a monograph on surgical knots and slings.^[1] Known under a variety of names, this knot has been used both on land and at sea.^[2] The common alternate name “lark’s head” is attributed to Tom Bowling (pseudonym) in the 1866 work The Book of Knots which is presumed to have been adapted from a French manuscript; lark’s head is a literal translation of the French name for the knot, tête d’alouette.^[3]

The underlying cow hitch structure can be formed and used in a variety of ways. These variations are differentiated by method used to form the knot and the way in which it is loaded.^[4] In particular, the knot can be formed with an end of the rope, in a closed loop or strap, or a combination of these two in which it is tied with the end and then formed into a loop by securing the free end to the standing part. Although certain names tend to be historically associated with a particular variations, real-world naming is not necessarily consistent between various users and applications.

• Halter hitch
• Hoxton knot
• List of knots
• Ringbolt hitching

Sprang is an ancient method of constructing fabric that has a natural elasticity. Its appearance is similar to netting, but unlike netting, sprang is constructed entirely from warp threads. Archaeological evidence indicates that sprang predates knitting; the two needlework forms bear a visible resemblance and serve similar functions but require different production techniques.

Although examples of sprang have been unearthed from as early as the Bronze Age, sprang was almost entirely undocumented in written records until the late nineteenth century when archaeological finds generated interest in Europe. Museum examples of sprang had been misidentified as knitting or lace until discoveries of ancient examples prompted reexamination of newer pieces. Subsequently, sprang has been identified in a variety of cultures and traditions across several continents. Its practice as a folk art waned during the twentieth century and sprang traditions have disappeared in most locales. Knitting has largely supplanted sprang.

In principle, sprang may be regarded as a practical application of the fiber manipulation techniques used in the children’s game known as cat’s cradle.^[1] Sprang is made by preparing a set of warp threads either on a rectangular frame or between a pair of beams. The crafter then generates a fabric by interlinking the warp threads. Unlike most textile production techniques that add new rows at the end of completed rows, sprang works upon the center of a group of fibers and the material grows inward from both ends with symmetrical top and bottom halves. The crafter must maintain control of the center area or the structure unravels: most sprang worked upon has too many fibers to keep in hand, so rods or sticks maintain the appropriate fiber positions. The fiber manipulation itself is done by the crafter’s fingers. Decorative patterns may be generated by regular variations in the fiber crossings. As crossings accumulate the crafter beats the material flat with a stick.^[2][3] Work continues with the material growing from both edges toward the middle until the center band becomes too tight to continue work.^[4] The completed fabric is either separated into two identical items or permanently secured at the middle to stabilize the contrary twists.^[3] In traditions where two halves of sprang are not severed by cutting, this center seam is a telltale sign of sprang manufacture.^[3]

While the word sprang is of Swedish origin, the technique has been practiced by many cultures around the world.^[3][5] The earliest surviving example of sprang is a hair net, c. 1400 BC, that was recovered from a bog in Denmark.^[2] Most archaeological finds of sprang fabric come from the later classical era and the start of the Early Medieval period: locations include Norway (third to fifth centuries AD), Switzerland, Egypt (possibly twenty-second dynasty, also early Coptic), and various Roman sites. Use of sprang has also been conjectured from archaeological recoveries of ancient looms and from depictions in period artwork.^[1]

Sprang is also an indigenous needlework technique among the peoples of South America, with the earliest known examples dating from before 900 AD among the Paracas culture and Nazca culture in present-day Peru.^[6] Recreation of archeological discoveries continue to inform about the presence and range of sprang in the Americas.^[7] Sprang has also been noted in the Middle East, Central Asia, the Indian subcontinent, and North America. Indigenous North American sprang includes woolen scarves by the Ho-Chunk of Wisconsin, and Hopi wedding sashes.^[5]

The natural elasticity of sprang makes it suitable for stockings, hair nets, sleeves, bags, scarves, and other purposes where pliant material is required.^[1][5] Most sprang needlework is utilitarian and hence was overlooked by scholars until late in its history, according to needlework historian Catherine Amoroso Leslie:^[5]

In fact, it was not until the nineteenth century and the discovery of sprang at archaeological sites that it was recognized as a separate and distinct form of needlework. Many museum objects that were wrongly classified as knitting or lace have now been correctly identified as sprang.

Following nearly simultaneous archaeological finds in Denmark and Egypt during the 1870s, an example of Coptic sprang brought to Austria in 1882 inspired the management of a factory near Vienna to manufacture sprang hammocks. The Paris World’s Fair of 1889 included an exhibit that duplicated ancient sprang, which led to renewed interest in the technique. Traditionally, cultures that practiced sprang had not produced written records of its manufacture or use. Shortly after the Paris exhibition, living traditions of sprang began to be documented. During the 1890s Ukrainian villagers were identified as practicing sprang in the manufacture of women’s caps. Soon afterward, local sprang production for various garments was also documented in parts of Eastern Europe and in Denmark. Local European traditions generally endured until the mid-twentieth century.^[5] Sprang is, however, largely a historical technique that has been supplanted for most purposes by the later invention of knitting, whose earliest known example dates from the third century AD.^[1]

During the late eighteenth and early nineteenth centuries a military fashion of decorative sprang sashes in silk known as faja in Spain gained international popularity among generals as a distinguishing ornament of rank. The fashion spread to northern Europe and to North America. George Washington wore a sash made of red silk sprang around the year 1779.^[5]

In a study of folk textiles of the Peloponnese in Greece, Linda Welters identifies the zonari as a long fringed belt constructed of sprang. The zonari is a rounded material of indeterminate length around 2 or 3 meters long and is wrapped multiple times when worn. From interviews conducted during the 1980s, Welters found that the technique was practiced by elderly women and regarded as a specialized skill which they worked on pairs of beams. Their vocabulary had no special term to correspond with sprang, so they referred to it with descriptive phrases such as knitting with sticks, plaiting, and weaving without passing through. The zonaria are made from sheep’s wool hand spun with a spindle and dyed red, blue, or black depending on local custom. Zonaria are a custom in much of Argolida and Corinthia where they are traditional dowry items worn by women from marriage onward and associated with fertility and donned on ritual occasions. Welters observes that these were mostly owned by older women who still wore folk clothing.^[3]

The sprang technique also survives as a traditional method for making hammocks in Guatemala, Mexico, and Colombia, and also in Colombia for making a shopping bag known as mechita. The work in these locations is generally performed on a backstrap loom. Handmade Mesoamerican sprang hammocks continue to be produced for export to a variety of international markets.^[2][5]

Additionally, sprang remains in use for making silk trouser drawstrings for male and female attire in Punjab in India and Pakistan. There, village women still use the technique to create colourful check/stipped/zigzag designs that have been handed down the generations. They finish the ends of the work using a weaving technique to make a hard 2-inch ‘harar’ with bunched twisted threads and knots. These sprang drawstrings are also used for the ‘ghugrai’ (full long skirts worn traditionally by Punjabi women) – it has been fashionable at times to have the drawstring ends on show, hanging with the ‘harar’ having colourful pom-poms or jewellery added on.
The sprang technique is also used to make the black ‘doria’ traditionally braided into women’s hair; they also have a colourful ‘harar’ ending fixed at the end. These can be found on sale in Punjab even now. Hairnets, baby cradles, and hanging baskets for fruit were also made. In attempts to retain traditional Punjabi culture, there are cultural craft centres in and around Jullunder where sprang technique is one of several old textile arts being passed on to young craftspeople.
Contemporary art textiles sometimes employ sprang in wall hangings.^[2]

In the 1970s new texts and research prompted makers to explore the craft.^[8] Interest in demonstrating and replicating sprang pieces^[9] has led to several books that revive the knowledge and expertise. Peter Collingwood’s Techniques of Sprang: Plaiting on Stretched Threads^[10] has provided foundational information that has influenced subsequent texts^[11].

• Regia Anglorum – Anglo-Saxon and Viking Crafts – Sprang
• Early Sprang Bibliography
• Hairnets and bonnets in Late Roman and Byzantine Egypt by Petra Linscheid
• Dutch website with English translation on sprang and nalbinding

In the mathematical field of knot theory, a split link is a link that has a (topological) 2-sphere in its complement separating one or more link components from the others.^[1] A split link is said to be splittable, and a link that is not split is called a non-split link or not splittable. Whether a link is split or non-split corresponds to whether the link complement is reducible or irreducible as a 3-manifold.

A link with an alternating diagram, i.e. an alternating link, will be non-split if and only if this diagram is connected. This is a result of the work of William Menasco.^[2] A split link has many connected, non-alternating link diagrams.

Cornrows (also called canerows in the Caribbean) are a style of three-strand braids in which the hair is braided very close to the scalp, using an underhand, upward motion to make a continuous, raised row.^[1][2] Cornrows are often done in simple, straight lines, as the term implies, but they can also be styled in elaborate geometric or curvilinear designs. They are considered a traditional hairstyle of African braiding practice in many African cultures, as well as in the African diaspora.^[3][4][5] They are distinct from, but may resemble, box braids, Dutch braids, melon coiffures, and other forms of plaited hair, and are typically tighter than braids used in other cultures.^[6]

The name cornrows or canerows refers to the layout of crops in corn and sugarcane fields in the Americas and Caribbean,^[1][7][2] where enslaved Africans were displaced during the Atlantic slave trade.^[8] According to Black folklore, cornrows were often used to communicate on the Underground Railroad and by Benkos Biohó during his time as a slave in Colombia.^[9] They often serve as a form of Black self-expression,^[10] especially among African Americans,^[1] but have been stigmatized in some cultures.^[5][11] Cornrows are traditionally called “kolese” or “irun didi” in Yoruba, and are often nicknamed “didi braids” in the Nigerian diaspora.^[12]

Cornrows are worn by both sexes, and are sometimes adorned with beads, shells, or hair cuffs.^[1] The duration of braiding cornrows may take up to five hours, depending on the quantity and width.^[13] Often favored for their easy maintenance, cornrows can be left in for weeks at a time if maintained through careful washing of the hair and natural oiling of the scalp. Braids are considered a protective styling on African curly hair as they allow for easy and restorative growth; braids pulled too tightly or worn for longer lengths of time and on different hair types can cause a type of hair loss known as traction alopecia.^[14]

Colonial attitudes and practices towards Black hairstyles reinforced racism, exclusion and inequality.^[52] For example, during the 18th century, slaves would sometimes have their hair shaved as a lesser form of punishment.^[15] Eurocentric beauty standards, which often denigrate Black hairstyles, can lead to internalized racism, colorism, and marginalization, which negatively affect Black people—and Black women in particular.^[47][53][54] Related valuations of hair texture—which portray straighter hair as “good hair” and curlier hair as “bad hair”—are emphasized through the media, advertising, and popular culture.^[53][55] These attitudes to hair can devalue African heritage and lead to discrimination.^[53][56] The unique type of discrimination that arises from prejudice towards Black women’s hair is called natural hair discrimination.^[57][58][59] Despite these challenges, cornrows have gained popularity among Black people as a way to express their Blackness, creativity and individuality.^[52][53][60]

Cornrows, alongside dreadlocks, have been the subject of several disputes in U.S. workplaces, as well as universities and schools. Some employers and educational institutions^[60] have considered cornrows unsuitable or “unprofessional”, and have banned them.^[48] Employees and civil rights groups have countered that such attitudes evidence cultural bias or racism, and some disputes have resulted in litigation.^[52][61] In 1981, Renee Rogers sued American Airlines for their policy which banned cornrows and other braided hairstyles. Other cases, such as Mitchell vs Marriott Hotel and Pitts vs. Wild Adventures, soon followed.^[62] Since other traditional Black hairstyles are also often banned, Black women may be forced to straighten their hair or emulate European hairstyles at significant additional cost.^[56] The intersection of racialized and gendered discrimination against Black women is often called misogynoir.^[63] In California, the CROWN Act was passed in 2019 to prohibit discrimination based on hair style and hair texture.^[64]

In 2011, the High Court of the United Kingdom, in a decision reported as a test case, ruled against a school’s decision to refuse entry to a student with cornrows. The school claimed this was part of its policy mandating “short back and sides” haircuts, and banning styles that might be worn as indicators of gang membership. However, the court ruled that the student was expressing a tradition and that such policies, while possibly justifiable in certain cases (e.g. skinhead gangs), had to accommodate reasonable racial diversities and cultural practices.^[65]

In some African nations, regularly changing hairstyles can be seen as a sign of social status for a woman, while advertising continues to promote straighter hairstyles as fashionable. Braids provide a way for women to maintain their hair, and are sometimes used with Chinese or Indian wigs to rotate hairstyles.^[55]

• 
  Styled cornrows
• 
  Cornrows
• 
  Artistic braiding

• List of hairstyles
• Shuku
• Koroba
• Crochet braids
• Waves (hairstyle)

• Media related to Cornrows at Wikimedia Commons
• The dictionary definition of cornrows at Wiktionary

In mathematics, the spherical braid group or Hurwitz braid group is a braid group on n strands. In comparison with the usual braid group, it has an additional group relation that comes from the strands being on the sphere. The group also has relations to the inverse Galois problem.^[1]

The spherical braid group on n strands, denoted ${\displaystyle SB_{n}}$ or ${\displaystyle B_{n}(S^{2})}$, is defined as the fundamental group of the configuration space of the sphere:^[2][3]
$${\displaystyle B_{n}(S^{2})=\pi _{1}(\mathrm {Conf} _{n}(S^{2})).}$$
The spherical braid group has a presentation in terms of generators ${\displaystyle \sigma _{1},\sigma _{2},\cdots ,\sigma _{n-1}}$ with the following relations:^[4]

• ${\displaystyle \sigma _{i}\sigma _{j}=\sigma _{j}\sigma _{i}}$ for ${\displaystyle |i-j|\geq 2}$
• ${\displaystyle \sigma _{i}\sigma _{i+1}\sigma _{i}=\sigma _{i+1}\sigma _{i}\sigma _{i+1}}$ for ${\displaystyle 1\leq i\leq n-2}$ (the Yang–Baxter equation)
• ${\displaystyle \sigma _{1}\sigma _{2}\cdots \sigma _{n-1}\sigma _{n-1}\sigma _{n-2}\cdots \sigma _{1}=1}$

The last relation distinguishes the group from the usual braid group.

Corned beef knot
Names	Corned beef knot, Salt pork knot
Category	Binding
Related	Packer’s knot
Releasing	Jamming
Typical use	cooking, baling, parcel tying
ABoK	#191
Instructions

The corned beef knot is a binding knot usually made in small line or string. It gains its name by often being used for binding the meat of the same name while it is being cooked. Since corned beef shrinks during cooking, the knot needs to be tightened several times during the process.

A buntline hitch is tied to the standing part and moderately tightened. The binding itself is tightened as the meat cooks by sliding the buntline hitch on the standing part. The knot is finished by a half hitch around the working end only after the meat has fully shrunk. It is considered more secure and suitable for this task than the related packer’s knot.^[1]

• List of binding knots
• List of knots

• http://notableknotindex.webs.com/butcherknots.html

Spanish Bowline
Category	Loop
Related	Portuguese bowline
Typical use	Lifting a person
ABoK	#1087

The Spanish bowline^[1] is a double loop knot that can be used to lift a person. For a conscious person, each loop is placed around a leg and the person holds onto the standing part of the rope. This knot can serve as a makeshift Bosun’s chair. For an unconscious person one loop is placed around the arm pits and the second loop is placed around the knees.

This knot has an elegant symmetry and can be tied quickly when mastered. Unlike the Portuguese bowline, each loop is separately fixed, which is a nice feature. It is a complicated knot however and if not properly tightened, it can slip. A fireman’s chair knot is another double loop knot which is more practical, easier to tie and less prone to slipping although it lacks the elegance of a Spanish bowline.

• 
  From behind
• 
  From the front

• List of knots

• Little Campfires (2020-08-01). “How to Tie a Bowline Knot (Plus Variations): Spanish Bowline”. YouTube.

In mathematical knot theory, a Conway sphere, named after John Horton Conway, is a 2-sphere intersecting a given knot or link in a 3-manifold transversely in four points. In a knot diagram, a Conway sphere can be represented by a simple closed curve crossing four points of the knot, the cross-section of the sphere; such a curve does not always exist for an arbitrary knot diagram of a knot with a Conway sphere, but it is always possible to choose a diagram for the knot in which the sphere can be depicted in this way.
A Conway sphere is essential if it is incompressible in the knot complement.^[1] Sometimes, this condition is included in the definition of Conway spheres.^[2]

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.

Rational tangles are equivalent if and only if their fractions are equal. An accessible proof of this fact is given in (Kauffman and Lambropoulou 2004). A number before an asterisk, *, denotes the polyhedron number; multiple asterisks indicate that multiple polyhedra of that number exist.^[2]

• Conway knot
• Dowker notation
• Alexander–Briggs notation
• Gauss notation

• Conway, J.H. (1970). “An Enumeration of Knots and Links, and Some of Their Algebraic Properties” (PDF). In Leech, J. (ed.). Computational Problems in Abstract Algebra. Pergamon Press. pp. 329–358. ISBN 0080129757.
• Kauffman, Louis H.; Lambropoulou, Sofia (2004). “On the classification of rational tangles”. Advances in Applied Mathematics. 33 (2): 199–237. arXiv:math/0311499. doi:10.1016/j.aam.2003.06.002. S2CID 119143716.