Author: Eggtimer

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. It is a knot invariant.

By way of example, the unknot has crossing number zero, the trefoil knot three and the figure-eight knot four. There are no other knots with a crossing number this low, and just two knots have crossing number five, but the number of knots with a particular crossing number increases rapidly as the crossing number increases.

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 (this sub-ordering is not based on anything in particular, except that torus knots then twist knots are listed first). The listing goes 3₁ (the trefoil knot), 4₁ (the figure-eight knot), 5₁, 5₂, 6₁, etc. This order has not changed significantly since P. G. Tait published a tabulation of knots in 1877.^[1]

There has been very little progress on understanding the behavior of crossing number under rudimentary operations on knots. A big open question asks 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, but this has not been proven.

Additivity of crossing number under knot sum has been proven for special cases, for example if the summands are alternating knots^[2] (or more generally, adequate knot), or if the summands are torus knots.^[3][4] Marc Lackenby has also given a proof that there is a constant N > 1 such that ⁠1/N⁠(cr(K₁) + cr(K₂)) ≤ cr(K₁ + K₂), but his method, which utilizes normal surfaces, cannot improve N to 1.^[5]

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 good predictor of the relative velocity of the DNA knot in agarose gel electrophoresis. Basically, the higher the crossing number, the faster the relative velocity. For composite knots, this does not appear to be the case, although experimental conditions can drastically change the results.^[6]

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.

Stevedore knot
Common name	Stevedore knot
Arf invariant	0
Braid length	7
Braid no.	4
Bridge no.	2
Crosscap no.	2
Crossing no.	6
Genus	1
Hyperbolic volume	3.16396
Stick no.	8
Unknotting no.	1
Conway notation	[42]
A–B notation	61
Dowker notation	4, 8, 12, 10, 2, 6
Last / Next	52 / 62
Other
alternating, hyperbolic, pretzel, prime, slice, reversible, tricolorable, twist

In knot theory, the stevedore knot is one of three prime knots with crossing number six, the others being the 6₂ knot and the 6₃ knot. The stevedore knot is listed as the 6₁ knot in the Alexander–Briggs notation, and it can also be described as a twist knot with four half twists, or as the (5,−1,−1) pretzel knot.

The mathematical stevedore knot is named after the common stevedore knot, which is often used as a stopper at the end of a rope. The mathematical version of the knot can be obtained from the common version by joining together the two loose ends of the rope, forming a knotted loop.

The stevedore knot is invertible but not amphichiral. Its Alexander polynomial is

${\displaystyle \Delta (t)=-2t+5-2t^{-1},\,}$

its Conway polynomial is

${\displaystyle \nabla (z)=1-2z^{2},\,}$

and its Jones polynomial is

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

The Alexander polynomial and Conway polynomial are the same as those for the knot 9₄₆, but the Jones polynomials for these two knots are different.^[2] Because the Alexander polynomial is not monic, the stevedore knot is not fibered.

The stevedore knot is a ribbon knot, and is therefore also a slice knot.

The stevedore knot is a hyperbolic knot, with its complement having a volume of approximately 3.16396.

• Figure-eight knot (mathematics)

In the mathematical field of knot theory, the crosscap number of a knot K is the minimum of

${\displaystyle C(K)\equiv 1-\chi (S),\,}$

taken over all compact, connected, non-orientable surfaces S bounding K; here ${\displaystyle \chi }$ is the Euler characteristic. The crosscap number of the unknot is zero, as the Euler characteristic of the disk is one.

The crosscap number of a knot sum is bounded:

${\displaystyle C(k_{1})+C(k_{2})-1\leq C(k_{1}\mathbin {\#} k_{2})\leq C(k_{1})+C(k_{2}).\,}$

• The crosscap number of the trefoil knot is 1, as it bounds a Möbius strip and is not trivial.
• The crosscap number of a torus knot was determined by M. Teragaito.

• Clark, B.E. “Crosscaps and Knots”, Int. J. Math and Math. Sci, Vol 1, 1978, pp 113–124
• Murakami, Hitoshi and Yasuhara, Akira. “Crosscap number of a knot,” Pacific J. Math. 171 (1995), no. 1, 261–273.
• Teragaito, Masakazu. “Crosscap numbers of torus knots,” Topology Appl. 138 (2004), no. 1–3, 219–238.
• Teragaito, Masakazu and Hirasawa, Mikami. “Crosscap numbers of 2-bridge knots,” Arxiv:math.GT/0504446.
• J.Uhing. “Zur Kreuzhaubenzahl von Knoten”, diploma thesis, 1997, University of Dortmund, (German language)

• “Crosscap Number“, KnotInfo.

Stevedore knot
Names	Stevedore knot, Double figure eight
Category	Stopper
Related	Figure-eight knot, Overhand knot, Figure-of-nine loop, Ashley’s stopper knot
Releasing	Non-jamming
Typical use	To provide a bulky, secure-when-slack stopper
ABoK	#456, #522[1]

The stevedore knot is a stopper knot, often tied near the end of a rope. It is more bulky and less prone to jamming than the closely related figure-eight knot.

The bight is given one more half turn than in the former knot [which itself is given, “one additional half twist,” more than the figure-eight knot], before the end is finally stuck. — The Ashley Book of Knots^[1]

In The Art of Knotting & Splicing, Cyrus Day explains that “the name originated in a pamphlet issued about 1890 by the C.W. Hunt Company, which sold rope under the name ‘Stevedore’. It was subsequently adopted by dictionaries, engineers’ handbooks, and other works of reference, and it is now firmly established in books, if not in the vocabulary of seamen.”^[2]

Despite this history, many sources, including The Ashley Book of Knots, presume that the knot was used by stevedores in their work loading and unloading ships.^[1]

The apparent aspect of the knot favored by transmission-line workers (to which the Hunt pamphlet was targeted) is the knot’s remaining easily untied after heavy loading. It should also be noted that the extra wrap that it has over the figure 8 stopper will, with flexible cordage, give better security when set.

The knot is formed by following the steps to make a figure-of-eight knot, but the working end makes an additional wrap around the standing part before passing back through the initial loop in the same direction it would have for a figure-of-eight knot (which yields a more secure & stable knot than were it to be a half-wrap less (a “Fig.9”)).

• List of knots
• Figure-of-nine loop

• Weisstein, Eric W. “Stevedores Knot”. MathWorld.

Crochet braids, also known as latch hook braids,^[1] are techniques for braiding hair that involve crocheting synthetic hair extensions to a person’s natural hair with a latch hook or crochet hook. While crochet braids are a hybrid of traditional braids, they’re considered to be more similar to weaves.^[2] This method is associated with African hair styles. Known as a protective style, the technique can assist with hair growth if cared for properly. Crochet braids can be worn straight, curly, twisted, or braided.^[3]

• List of hairstyles

Stafford knot
The Stafford badge
Information
Family	Stafford family

The Stafford knot, more commonly known as the Staffordshire knot, is a distinctive three-looped knot that is the traditional symbol of the English county of Staffordshire and of its county town, Stafford. It is a particular representation of the simple overhand knot, the most basic knot of all.^[1][2]

One legend of its origin, generally considered mythical, is that three convicted criminals who had committed a crime together were due to be executed in Stafford gaol. There was argument over who should be hanged first but the hangman solved the problem by devising this knot and hanging the three simultaneously.^[1] The knot can be seen on a 4-foot-high (1.2 m) carved Anglo-Saxon cross in a churchyard in Stoke-upon-Trent, giving its name as the Staffordshire Knot and also on a seventh-century Anglo-Saxon object from the Staffordshire hoard.^[3] This strongly suggests it pre-dates the Norman and medieval period, being probably either a heraldic symbol of early Mercia or a Celtic Christian symbol brought to Staffordshire by missionary monks from Lindisfarne.

The flag of the historic county of Staffordshire

The Stafford family heraldic flag, blazoned with eleven Stafford knots

Flag of the city of Brisbane

The earliest known appearance of the knot in association with the Stafford family was on the 15th-century seal of Lady Joan de Stafford, Lady Wake (daughter of Hugh de Stafford, 2nd Earl of Stafford), who adapted the Wake knot for her use. The seal, on which four knots appear tied on a string around her coat of arms, is now in the British Museum.^[4][5] After Lady Wake died childless in 1443, her nephew Humphrey de Stafford, 6th Earl of Stafford inherited her personal property. Humphrey, who was made the Duke of Buckingham the following year, incorporated his aunt’s adaptation of the Wake knot as a cordon around his seal, although he used three knots and not four.^[4]

The knot appears on a drawing of the standard of Sir Henry de Stafford, which was flown c. 1475. In a visitation of Stafford in 1583, the town’s arms were recorded for the first time as or a chevron gules, a Stafford knot argent. These arms, which represent the traditional Stafford arms with the addition of the knot, is still the flag of Staffordshire (albeit with the knot changed from white to gold).^[4][1][6]

Future earls of Stafford retained the knot for use on heraldic badges, although the exact date of use is unknown. Early heraldic badges are poorly recorded, however, and many had multiple badges. In 1720, the Earl of Stafford used 18 badges, including the Stafford knot, all enclosed in a circle of Stafford knots.^[2][4] The design closely matches the early design of the pretzel, which was made to represent arms crossed in prayer. In the 19th century, it became a symbol of Staffordshire pottery.^[7] This “Staffordshire knot” was also used as a surgical suture in the 19th century. It was discarded by some as too dangerous, as the knot could slip if not tied correctly, and indeed, its use led to multiple deaths by haemorrhage.^[8][9]

The knot appears prominently on the flag of the historic county of Staffordshire, as well as on the banners of Staffordshire County Council and Staffordshire Moorlands District Council. The motto of the county council – “the knot unites” – refers to the Stafford knot. It also appears on the coat of arms of the Borough of Stafford.^[1] The knot appears on the logos of both Staffordshire University and Keele University, as well as on many school badges.

Stoke-on-Trent based professional football team Port Vale use the knot on their club badge, as do fellow Staffordshire clubs Chasetown, Hanley Town, Hednesford Town, Leek Town, Lichfield City, Newcastle Town, Tamworth, Stafford Rangers and Wolverhampton Casuals. Additionally, West Midlands club Darlaston Town (1874) also feature the knot as the town of Darlaston was historically part of Staffordshire. The knot also features on the badge of Staffordshire County Cricket Club.

The Stafford knot was the badge of the 38th (1st Staffordshire) Regiment of Foot in which Sir Thomas Brisbane entered the British Army as an Ensign in 1789 – and thus the eponymous city of Brisbane in Queensland, Australia, has the Stafford knot on its arms. The knot formed part of the insignia of the North Staffordshire Regiment, South Staffordshire Regiment, Staffordshire Regiment and currently is the arm badge of the Mercian Regiment. It is also used as the badge of the Staffordshire Police force and Staffordshire Fire and Rescue Service. Staffordshire wing of the Air Training Corps also uses the knot on its crest.

Also used as a pin/lapel badge by members of Freemasonry under the Provincial Grand Lodge of Staffordshire.

The North Staffordshire Railway, an independent railway company in the county from 1845 until the compulsory grouping of the railways in 1923, was colloquially referred to as The Knotty after the knot. The knot symbol appeared in the railway company’s badge. The Smiler,^[10] a roller coaster at Alton Towers, has a combined section of track known as the “Stafford/Staffordshire Knot”, consisting of a cobra roll entwined with a batwing. It is given this nickname due to the resemblance to the original knot shape.

In an article published in the Birmingham Evening Mail on 1 June 1965, journalist Roy Smith described plans for the Gravelly Hill Interchange as “like a cross between a plate of spaghetti and an unsuccessful attempt at a Staffordshire knot”, with the headline above the article on the newspaper’s front page, written by sub-editor Alan Eaglesfield, reading “Spaghetti Junction”.^[11][12][13]

Crochet (English: /kroʊˈʃeɪ/;^[1] French: [kʁɔʃɛ]^[2]) is a process of creating textiles by using a crochet hook to interlock loops of yarn, thread, or strands of other materials.^[3] The name is derived from the French term crochet, which means ‘hook’.^[4] Hooks can be made from different materials (aluminum, steel, metal, wood, bamboo, bone, etc.), sizes, and types (in-line, tapered, ergonomic, etc.). The key difference between crochet and knitting, beyond the implements used for their production, is that each stitch in crochet is completed before the next one, while knitting keeps many stitches open at a time. Some variant forms of crochet however, such as Tunisian crochet and Broomstick lace, do keep multiple crochet stitches open at a time.

The word crochet is derived from the French word crochet, a diminutive of croche, in turn from the Germanic croc, both meaning “hook”.^[3] It was used in 17th-century French lace-making, where the term crochetage designated a stitch used to join separate pieces of lace. The word crochet subsequently came to describe both the specific type of textile and the hooked needle used to produce it.^[5]

In 1567, the tailor of Mary, Queen of Scots, Jehan de Compiegne, supplied her with silk thread for sewing and crochet, “soye à coudre et crochetz“.^[6]

Knitted textiles survive from as early as the 11th century CE, but the first substantive evidence of crocheted fabric emerges in Europe during the 19th century.^[7] Prior to the 19th century, classifications often failed to distinguish between crochet and knitting.^[8] Earlier work identified as crochet was commonly made by nålebinding, a different looped yarn technique. There are three main theories for the origin of crochet.^[9] Some believe that its beginnings can be traced to the Arab trade route, originating in Arabia and spreading to Tibet and then Spain as well as other Mediterranean countries.^[9] The second theory places crochet in South America where it was used as adornment in a tribe’s puberty ritual.^[9] The third notes the use of crochet in China where early examples of dolls were worked entirely in crochet.^[9]

The first known published instructions for crochet explicitly using that term to describe the craft in its present sense appeared in the Dutch magazine Penélopé in 1823. This includes a color plate showing five styles of purse, of which three were intended to be crocheted with silk thread.^[10] The first is “simple open crochet” (crochet simple ajour), a mesh of chain-stitch arches. The second (illustrated here) starts in a semi-open form (demi jour), where chain-stitch arches alternate with equally long segments of slip-stitch crochet, and closes with a star made with “double-crochet stitches” (dubbelde hekelsteek: double-crochet in international terminology; single-crochet in US).^[11] The third purse is made entirely in double-crochet. The instructions prescribe the use of a tambour needle (as illustrated below) and introduce a number of decorative techniques.

The earliest dated reference in English to garments made of cloth produced by looping yarn with a hook—shepherd’s knitting—is in The Memoirs of a Highland Lady by Elizabeth Grant (1797–1830). The journal entry, itself, is dated 1812 but was not recorded in its subsequently published form until some time between 1845 and 1867, and the actual date of publication was first in 1898.^[12] Nonetheless, the 1833 volume of Penélopé describes and illustrates a shepherd’s hook, and recommends its use for crochet with coarser yarn.^[13]

In 1844, one of the numerous books discussing crochet that began to appear in the 1840s states:

Crochet needles, sometimes called Shepherds’ hooks, are made of steel, ivory, or box-wood. They have a hook at one end similar in shape to a fish-hook, by which the wool or silk is caught and drawn through the work. These instruments are to be procured of various sizes…^[14]

Two years later, the same author writes:

Crochet, — a species of knitting originally practised by the peasants in Scotland, with a small hooked needle called a shepherd’s hook, — has, within the last seven years, aided by taste and fashion, obtained the preference over all other ornamental works of a similar nature. It derives its present name from the French; the instrument with which it is worked being by them, from its crooked shape, termed ‘crochet.’ This art has attained its highest degree of perfection in England, whence it has been transplanted to France and Germany, and both countries, although unjustifiably, have claimed the invention.^[15]

An instruction book from 1846 describes Shepherd or single crochet as what in current international terminology is either called single crochet or slip-stitch crochet, with U.S. terminology always using the latter (reserving single crochet for use as noted above).^[16] It similarly equates “Double” and “French crochet”.^[17]

Notwithstanding the categorical assertion of a purely British origin, there is solid evidence of a connection between French tambour embroidery, french passementerie and crochet. A form of hook known as crochet was used to create ‘chains in the air’ as part of passementerie back in the 17th century. This is confirmed by a patent issued to the passementiers by Louis XIV in 1653, and there are earlier decorative examples of this technique. The patent lists various items, including “thread for embroidery, enhanced and embellished as done with a needle, on thimbles, on the fingers, on a crochet, and on a bobbin.” Similarly, chain stitch appears in Queen Elizabeth I’s wardrobe accounts, starting in 1558, with further references to garments bordered with ‘cheyne lace’ in other inventories. One example from 1588 describes “a long cloak of murry velvet, with a border of small cheyne lace of Venice silver.” While the exact design of the 1653 crochet is unclear, a 1723 French dictionary by Jacques Savary des Brûlons describes a crochet as a small iron instrument, three or four inches long, with a pointed, curved end and a wooden handle, used by passementiers for tasks like creating hat seams and attaching flowers to mesh. It is most likely that the hook used in crochet came from the ones used by the French pessamenterie industry.^[18]
French tambour embroidery and the crochet needle used for it was illustrated in detail in 1763 in Diderot’s Encyclopedia. The tip of the needle shown there is indistinguishable from that of a present-day inline crochet hook and the chain stitch separated from a cloth support is a fundamental element of the latter technique. The 1823 Penélopé instructions unequivocally state that the tambour tool was used for crochet and the first of the 1840s instruction books uses the terms tambour and crochet as synonyms.^[19] This equivalence is retained in the 4th edition of that work, 1847.^[20]

The strong taper of the shepherd’s hook eases the production of slip-stitch crochet but is less amenable to stitches that require multiple loops on the hook at the same time. Early yarn hooks were also continuously tapered but gradually enough to accommodate multiple loops. The design with a cylindrical shaft that is commonplace today was largely reserved for tambour-style steel needles. Both types gradually merged into the modern form that appeared toward the end of the 19th century, including both tapered and cylindrical segments, and the continuously tapered bone hook remained in industrial production until World War II.

The early instruction books make frequent reference to the alternative use of ‘ivory, bone, or wooden hooks’ and ‘steel needles in a handle’, as appropriate to the stitch being made. Taken with the synonymous labeling of shepherd’s- and single crochet, and the similar equivalence of French- and double crochet, there is a strong suggestion that crochet is rooted both in tambour embroidery and shepherd’s knitting, leading to thread and yarn crochet respectively; a distinction that is still made. The locus of the fusion of all these elements—the “invention” noted above—has yet to be determined, as does the origin of shepherd’s knitting.

Shepherd’s hooks are still being made for local slip-stitch crochet traditions. The form in the accompanying photograph is typical for contemporary production. A longer continuously tapering design intermediate between it and the 19th-century tapered hook was also in earlier production, commonly being made from the handles of forks and spoons.

In the 19th century, as Ireland was facing the Great Irish Famine (1845–1849), crochet lace work was introduced as a form of famine relief^[21] (the production of crocheted lace being an alternative way of making money for impoverished Irish workers).^[22] Men, women, children joined a co-operative in order to crochet and produce products to help with famine relief during the Great Irish Famine. Schools to teach crocheting were started. Teachers were trained and sent across Ireland to teach this craft. When the Irish immigrated to the Americas, they were able to take with them crocheting.^[23] Mademoiselle Riego de la Branchardiere is generally credited with the invention of Irish Crochet, publishing the first book of patterns in 1846. Irish lace became popular in Europe and America, and was made in quantity until the first World War.^[24]

Fashions in crochet changed with the end of the Victorian era in the 1890s. Crocheted laces in the new Edwardian era, peaking between 1910 and 1920, became even more elaborate in texture and complicated stitching.

The strong Victorian colors disappeared, though, and new publications called for white or pale threads, except for fancy purses, which were often crocheted of brightly colored silk and elaborately beaded. After World War I, far fewer crochet patterns were published, and most of them were simplified versions of the early 20th-century patterns. After World War II, from the late 1940s until the early 1960s, there was a resurgence in interest in home crafts, particularly in the United States, with many new and imaginative crochet designs published for colorful doilies, potholders, and other home items, along with updates of earlier publications. These commercial pattern books, such as the Kentucky Rug Yarns series, published by the January and Wood company, provided simplified instructions for practical household items.^[25] These patterns called for thicker threads and yarns than in earlier patterns and included variegated colors. The craft remained primarily a homemaker’s art until the late 1960s and early 1970s, when the new generation picked up on crochet and popularized granny squares, a motif worked in the round and incorporating bright colors.

Although crochet underwent a subsequent decline in popularity, the early 21st century has seen a revival of interest in handcrafts and DIY, as well as improvement of the quality and varieties of yarn. As well as books and classes, there are YouTube tutorials and TikTok videos to help people who may need a clearer explanation to learn how to crochet.^[26] Crochet has been used as a medium to explore identity and self-expression in educational programs, bridging the gap between art and technology. The repetitive nature of the craft is also used as a therapeutic tool for stress relief, depression, and anxiety.^[27]

The COVID-19 pandemic has contributed to the rise of crocheting, since people were staying inside more and wanted to pick up new hobbies/activities. Crochet garments have also been popular recently, as it was worn by numerous pop culture icons like Harry Styles, Dua Lipa, and featured on the Netflix show Wednesday.  Crochet has experienced a revival on the catwalk as well. Christopher Kane’s Fall 2011 Ready-to-Wear collection.^[28] Additionally, it was featured in luxury brands such as Moschino, who has crocheted skirts in their Ready-to-Wear Spring 2024 collection, and Miumiu, who have designed and sold several crocheted items from their brand such as a crocheted scrunchie. Online searches for crochet items have rose drastically with the trend skyrocketing, with a 32% rise in searches for the keyword ‘crochet’ on online clothing site Depop.^[29] With the trend of crochet being on the rise in social media, fast fashion brands like Shein have been trying to create products that resemble crochet items.

Today, many popular crochet projects amongst youth include creating stuffed animals, hats, and even flowers. Websites such as Etsy and Ravelry have made it easier for individual hobbyists to sell and distribute their patterns or projects across the internet. Microentrepreneurs have used social media to share their designs and grow their business. They use social media accounts as a tool to measure their products popularity, stay relevant, and develop relationships with customers. A passive behavior such as liking or sharing will show up on a person’s social media page which will encourage watchers to follow, view videos, and use hashtags related to that content for easier access in the future.^[30]

Micro crochet, Filet crochet, Tunisian crochet, tapestry crochet, broomstick lace, hairpin lace, cro-hooking, and Irish crochet are all variants of the basic crochet method.

Sustainability can be a key pillar of crochet, as it may use upcycled and environmentally friendly materials in yarn to create sustainable fabrics such as bamboo, hemp, cotton, linen, and wool. Since crochet is typically made by hand, although there are crochet machines^[31] in the process of being developed, crochet contributes to ethical production by adhering to the slow production of garments and accessories rather than the fast-paced and mechanized garments that are produced on a mass scale by the fast fashion industry.

Slow fashion is part of the “slow movement”. The slow movement’s objective in clothing and garment manufacturing is to promote respect for people, animals and the environment. Slow fashion supports local artists, the use of eco-friendly materials, and preserves craftsmanship.^[32] Since slow fashion promotes the skill of being able to crochet, it’s able to provide jobs with a fair income for unemployed youths in Nigeria. Money created from selling the crochet products is a method to improve their standard of living and meet sustainable development goal 8 (SDG).^[33]

Slow fashion emphasizes the importance of providing high quality products over quantity, resulting in consumers being more aware of the ethics and process of material production and labor.^[32] A project called ‘Mini Fashion Statement’ was created in order to outline the condition of the factories/provide their opinions on the ‘true cost’ of fashion. Then these scrolls were placed in garments of fast fashion brands. The goal of this project was to raise awareness of ethical implications in the supply chain, which encourages consumers, who often purchase fast fashion, to rethink what they buy. (Corbett, 2019).  Additionally, some people choose to crochet their own luxury items that are inspired by luxury fashion. The counterfeit crochet project was founded in 2006. Its goal was to bring to light that luxury can only be accessed by certain people. This project’s goal was to encourage people to use their crochet skills to replicate luxury fashion like Chanel and Louis Vuitton rather than buying from those companies.^[34] Additionally, according to Kate Fletcher, slow fashion encourages using eco-friendly materials to minimize harm on the planet and to focus more on creating higher quality goods which in turn allows people to recognize the quality of different sustainable yarn such as natural fibers as well as recycled yarns. Additionally places like Milan offer markets where vendors from other regions come to sell yarn. Since the material comes locally the need to transport raw materials a long distance is reduced therefore reducing carbon emissions. Regional manufacturers also produce items closer to the point of consumption as it increases personal connection between consumer and producer, reduces carbon emission, and provides jobs.^[32]

In addition, crochet fabric can be loosened and undone easily to be reused in a new way. Crochet can be added to worn or torn garments and to add a stylistic enhancement, such as a collar or sleeves, which may contribute to upcycling of clothing. Crochet highlights the sustainability factor of quality versus quantity because of the customization aspect which can make crochet items more personalized. Another aspect of the sustainability of crochet is that crochet is versatile and can be used to create many different products eliminating the need to constantly buy new products.  ^[35]

Synthetic fibers and mixed fibers such as acrylic, polyester and nylon are quite commonly used. Synthetic fibers are not biodegradable and aren’t recyclable. Today, numerous garments, especially fast fashion, are made from synthetic fibers or mixed fibers. The amount of synthetic fibers used has increased significantly over the years. In 1990 natural fibers made up 60% of fiber consumption and in 2000 the number declined to 45%, with the number declining to 40% 8 years later. In 2009, 35.8 million tons of synthetic fibers were used and 23.3 million tons of cotton were used.^[36] In 2023, the amount of total fibers produced and used was 125 million fibers, and it increased to 132 million tons in 2024. The most commonly produced global synthetic fiber is polyester, making up 59%, with 88% of it being fossil based. If this trend continues, then the expected amount produced in 2030 is 169 million tons.^[37] This has more than doubled in the amount produced in 2009.

Many crochet and knit yarn brands have taken a stance on sustainability by aiming to increase the production of natural fibers such as organic cotton, hemp, wool and recycled yarns.

Creating crocheted items has become a way to make sustainable fashion.^[38] Fast fashion brands like Shein^[39] have created products that resemble crocheted items. Additionally, The Crochet-Style Button-Through Sweater Polo, by Abercrombie and Fitch resemble designs popular in the 1960s and ’70s. Although the pattern resembles crochet in reality it’s made from a knit machine. These shirts are made from a combination of cotton, acrylic or viscose, which are a typical fabrication for fast-fashion pieces. Although Abercrombie may not consider itself fast fashion, there is little to suggest that its materials and design practices differ from Shein or Zara, other than the quantity of products made.

The basic materials required for crocheting are a hook, scissors (to cut yarn), and some type of material that will be crocheted, the most commonly used are yarn or thread. Alternatively, some people choose to crochet with their hands, especially for large yarns. Yarn, one of the most commonly used materials for crocheting, has varying weights which need to be taken into consideration when following patterns. The weight of the yarn can affect not only the look of the product but also the feeling. Acrylic can also be used when crocheting, as it is synthetic and an alternative to wool. Additional tools are convenient for making related accessories. Examples of such tools include cardboard cutouts, which can be used to make tassels, fringe, and many other items; a pom-pom circle, used to make pom-poms; a tape measure and a gauge measure, both used for measuring crocheted work and counting stitches; a row counter; and occasionally plastic rings, which are used for special projects. In recent years, yarn selections have moved beyond synthetic and plant and animal-based fibers to include bamboo, qiviut, hemp, and banana stalks, to name a few. Many advanced crocheters have also incorporated recycled materials into their work in an effort to “go green” and experiment with new textures by using items such as plastic bags, old T-shirts or sheets, VCR or cassette tape, and ribbon.^[40]

Crochet hook types

The crochet hook comes in many sizes and materials. Because sizing is categorized by the diameter of the hook’s shaft, a crafter aims to create stitches of a certain size in order to reach a particular gauge specified in a given pattern. If gauge is not reached with one hook, another is used until the stitches made are the needed size. Crafters may have a preference for one type of hook material over another due to aesthetic appeal, yarn glide, or hand disorders such as arthritis, where bamboo or wood hooks are favored over metal for the perceived warmth and flexibility during use. Hook grips and ergonomic hook handles are also available to assist crafters.

Aluminum, bamboo, and plastic crochet hooks are available from 2.25 to 30 millimeters in size, or from B-1 to T/X in American sizing.^[41] Artisan-made hooks are often made of hand-turned woods, sometimes decorated with semi-precious stones or beads.

Steel crochet hooks are sized in a reverse manner – the higher the number, the smaller the hook. They range in size from 0.9 to 2.7 millimeters, or from 14 to 00 in American sizing.^[41] These hooks are used for fine crochet work such as doilies and lace.

Crochet hooks used for Tunisian crochet are elongated and have a stopper at the end of the handle, while double-ended crochet hooks have a hook on both ends of the handle. Tunisian crochet hooks are shaped without a fat thumb grip and thus can hold many loops on the hook at a time without stretching some to different heights than others (Solovan). There is also a double hooked tool called a Cro-hook. While this is not in itself a hook, it is a device used in conjunction with a crochet hook to produce stitches.

Yarn types

Yarn for crochet is usually sold as balls, or skeins (hanks), although it may also be wound on spools or cones. Skeins and balls are generally sold with a yarn band, a label that describes the yarn’s weight, length, dye lot, fiber content, washing instructions, suggested needle size, likely gauge, etc. It is a common practice to save the yarn band for future reference, especially if additional skeins must be purchased. Crocheters generally ensure that the yarn for a project comes from a single dye lot. The dye lot specifies a group of skeins that were dyed together and thus have precisely the same color; skeins from different dye lots, even if very similar in color, are usually slightly different and may produce a visible stripe when added onto existing work. If insufficient yarn of a single dye lot is bought to complete a project, additional skeins of the same dye lot can sometimes be obtained from other yarn stores or online.

The thickness or weight of the yarn is a significant factor in determining how many stitches and rows are required to cover a given area for a given stitch pattern. This is also termed the gauge. Thicker yarns generally require large-diameter crochet hooks, whereas thinner yarns may be crocheted with thick or thin hooks. Hence, thicker yarns generally require fewer stitches, and therefore less time, to work up a given project. The recommended gauge for a given ball of yarn can be found on the label that surrounds the skein when buying in stores. Patterns and motifs are coarser with thicker yarns and produce bold visual effects, whereas thinner yarns are best for refined or delicate pattern-work. Yarns are standardly grouped by thickness into six categories: superfine, fine, light, medium, bulky and superbulky. Quantitatively, thickness is measured by the number of wraps per inch (WPI). The related weight per unit length is usually measured in tex or denier.

Before use, hanks are wound into balls in which the yarn emerges from the center, making crocheting easier by preventing the yarn from becoming easily tangled. The winding process may be performed by hand or done with a ball winder and swift.^[42]

A yarn’s usefulness is judged by several factors, such as its loft (its ability to trap air), its resilience (elasticity under tension), its washability and colorfastness, its hand (its feel, particularly softness vs. scratchiness), its durability against abrasion, its resistance to pilling, its hairiness (fuzziness), its tendency to twist or untwist, its overall weight and drape, its blocking and felting qualities, its comfort (breathability, moisture absorption, wicking properties) and its appearance, which includes its color, sheen, smoothness and ornamental features. Other factors include allergenicity, speed of drying, resistance to chemicals, moths, and mildew, melting point and flammability, retention of static electricity, and the propensity to accept dyes. Desirable properties may vary for different projects, so there is no one “best” yarn.

Although crochet may be done with ribbons, metal wire, or more exotic filaments, most yarns are made by spinning fibers. In spinning, the fibers are twisted so that the yarn resists breaking under tension; the twisting may be done in either direction, resulting in a Z-twist or S-twist yarn. Due to the counter-clockwise action of the crochet stitch, a Z-plied yarn is most desirable for crochet.^[43] If the fibers are first aligned by combing them and the spinner uses a worsted type drafting method such as the short forward draw, the yarn is smoother and called a worsted; by contrast, if the fibers are carded but not combed and the spinner uses a woolen drafting method such as the long backward draw, the yarn is fuzzier and called woolen-spun. The fibers making up a yarn may be continuous filament fibers such as silk and many synthetics, or they may be staples (fibers of an average length, typically a few inches); naturally filament fibers are sometimes cut up into staples before spinning. The strength of the spun yarn against breaking is determined by the coarseness and tensile strength of the raw material, the amount of twist, the length of the fibers, and the thickness of the yarn. In general, yarns become stronger with more twists (also called worst), longer fibers, and thicker yarns (more fibers); for example, thinner yarns require more twists than thicker yarns to resist breaking under tension. The thickness of the yarn may vary along its length; a slub is a much thicker section in which a mass of fibers is incorporated into the yarn.^[44]

The spun fibers are generally divided into animal fibers, plant, and synthetic fibers. These fiber types are chemically different, corresponding to proteins, carbohydrates, and synthetic polymers, respectively. Animal fibers include silk, but generally are long hairs of animals such as sheep (wool), goat (angora, or cashmere goat), rabbit (angora), llama, alpaca, dog, cat, camel, yak, and muskox (qiviut). Plants used for fibers include cotton, flax (for linen), bamboo, ramie, hemp, jute, nettle, raffia, yucca, coconut husk, banana trees, soy and corn. Rayon and acetate fibers are also produced from cellulose mainly derived from trees. Common synthetic fibers include acrylics,^[45] polyesters such as dacron and ingeo, nylon and other polyamides, and olefins such as polypropylene. Of these types, wool is generally favored for crochet, chiefly owing to its superior elasticity, warmth and (sometimes) felting; however, wool is generally less convenient to clean and some people are allergic to it. It is also common to blend different fibers in the yarn, e.g., 85% alpaca and 15% silk. Even within a type of fiber, there can be great variety in the length and thickness of the fibers; for example, Merino wool and Egyptian cotton are favored because they produce exceptionally long, thin (fine) fibers for their type.

A single spun yarn may be crochet as is, or braided or plied with another. In plying, two or more singles are spun together, almost always in the opposite sense from which they were spun individually; for example, two Z-twist yarns are usually plied with an S-twist. The opposing twist relieves some of the yarn’s tendency to curl up and produces a thicker, balanced yarn. Plied yarns may themselves be plied together, producing cabled yarns or multi-stranded yarns. Sometimes, the yarns being plied are fed at different rates, so that one yarn loops around the other, as in bouclé. The single yarns may be dyed separately before plying, or afterward to give the yarn a uniform look.

The dyeing of yarns is a complex art. Yarns need not be dyed, or they may be dyed one color or a great variety of colors. Dyeing may be done industrially, by hand, or even hand-painted onto the yarn. A great variety of synthetic dyes have been developed since the synthesis of indigo dye in the mid-19th century; however, natural dyes are also possible, although they are generally less brilliant. The color scheme of a yarn is sometimes called its colorway. Variegated yarns can produce interesting visual effects, such as diagonal stripes.

Crocheted fabric is begun by placing a slip-knot loop on the hook (though other methods, such as a magic ring or simple folding over of the yarn, may be used), pulling another loop through the first loop, and repeating this process to create a chain of a suitable length. The chain is either turned and worked in rows, or joined to the beginning of the row with a slip stitch and worked in rounds. Rounds can also be created by working many stitches into a single loop. Stitches are made by pulling one or more loops through each loop of the chain. At any one time at the end of a stitch, there is only one loop left on the hook. Tunisian crochet, however, draws all of the loops for an entire row onto a long hook before working them off one at a time. Like knitting, crochet can be worked either flat (back and forth in rows) or in the round (in spirals, such as when making tubular pieces).

There are six main types of basic stitches (the following description uses international crochet terminology with US variants noted in brackets).

1. Chain stitch (ch) – the most basic of all stitches and used to begin most projects. Yarn round hook (yrh) and draw through.
2. Slip stitch (sl st or ss) – used to join chain stitch to form a ring. Insert hook in work, yrh, draw through.
3. Double crochet (dc) (US = single crochet) – Insert hook, draw loop through, (2 loops on hook, hence double), yrh, draw through.
4. Half treble (htr) (US = half double) – yrh, insert hook, draw loop through, (3 loops on hook, hence treble), yrh, draw through all loops.
5. Treble (tr) (US = double) – yrh, insert hook, draw loop through (3 loops on hook, hence treble), yrh, draw through 2 loops, yrh, draw through 2 loops.
6. Double treble (US = treble or triple) – as treble but 2 yrh at start (hence double treble). Also triple treble (ttr), as treble but with 3 yrh at start, and so on.

While the horizontal distance covered by these basic stitches is the same, they differ in height and can be replaced with a length of ch when required, e.g. 1 tr = 3 ch.^[46]

The more advanced stitches are often combinations of these basic stitches, or are made by inserting the hook into the work in unusual locations. More advanced stitches include the shell stitch, V stitch, spike stitch, Afghan stitch, butterfly stitch, popcorn stitch, cluster stitch, and crocodile stitch.

There are two main notations of basic stitches, one used across Europe, Australia, India and other crocheting nations, the other in the US and Canada. (In America, international terminology is often erroneously called British or UK terminology.)
Crochet is traditionally worked from a written pattern using standard abbreviations or from a diagram, thus enabling non-English speakers to use English-based patterns.^[47] To help counter confusion when reading patterns, a diagramming system using a standard international notation has come into use (illustration, left). In the United States, crochet terminology and sizing guidelines, as well as standards for yarn and hook labeling, are primarily regulated by the Craft Yarn Council.^[48]

Another terminological difference is known as tension (international) and gauge (US). Individual crocheters work yarn with a loose or a tight hold and, if unmeasured, these differences can lead to significant size changes in finished garments that have the same number of stitches. In order to control for this inconsistency, printed crochet instructions include a standard for the number of stitches across a standard swatch of fabric. An individual crocheter begins work by producing a test swatch and compensating for any discrepancy by changing to a smaller or larger hook.^[50][51]

One of the more obvious differences is that crochet uses one hook while most knitting uses two needles. In most crochet, the artisan usually has only one live stitch on the hook (except Tunisian crochet), while a knitter keeps an entire row of stitches active simultaneously. Dropped stitches, which can unravel a knitted fabric, rarely interfere with crochet work, due to a second structural difference between knitting and crochet. In knitting, each stitch is supported by the corresponding stitch in the row above and it supports the corresponding stitch in the row below. In contrast, crochet stitches are only supported by and support the stitches on either side of it. If a stitch in a finished crocheted item breaks, the stitches above and below remain intact, and because of the complex looping of each stitch, the stitches on either side are unlikely to come loose unless heavily stressed^[52]
Round or cylindrical patterns are simple to produce with a regular crochet hook, but cylindrical knitting requires either a set of circular needles or three to five special double-ended needles.

Freeform crochet is a technique that can create interesting shapes in three dimensions because new stitches can be made independently of previous stitches almost anywhere in the crocheted piece. It is generally accomplished by building shapes or structural elements onto existing crocheted fabric at any place the crafter desires.

Knitting can be accomplished by machine, while many crochet stitches can only be crafted by hand. The height of knitted and crocheted stitches is also different: a single crochet stitch is twice the height of a knit stitch in the same yarn size and comparable diameter tools, and a double crochet stitch is about four times the height of a knit stitch.^[53]

While most crochet is made with a hook, there is also a method of crocheting with a knitting loom. This is called loomchet.^[54] Slip stitch crochet is very similar to knitting. Each stitch in slip stitch crochet is formed the same way as a knit or purl stitch which is then bound off. A person working in slip stitch crochet can follow a knitted pattern with knits, purls, and cables, and get a similar result.^[55]

It is a common perception that crochet produces a thicker fabric than knitting, tends to have less “give” than knitted fabric, and uses approximately a third more yarn for a comparable project than knitted items. Although this is true when comparing a single crochet swatch with a stockinette swatch, both made with the same size yarn and needle/hook, it is not necessarily true for crochet in general. Most crochet uses far less than 1/3 more yarn than knitting for comparable pieces, and a crocheter can get similar feel and drape to knitting by using a larger hook or thinner yarn. According to sources^[56] claiming to have tested the 1/3 more yarn assertion, a single crochet stitch (sc) uses approximately the same amount of yarn as knit garter stitch, but more yarn than stockinette stitch. Any stitch using yarnovers uses less yarn than single crochet to produce the same amount of fabric. Stitches which are in fact multiple stitches worked together, such as cluster stitches, bobble stitches, and puff stitches, will use the most length.^[57]

Standard crochet stitches like sc and dc also produce a thicker fabric, more like knit garter stitch. This is part of why they use more yarn. Slip stitch can produce a fabric much like stockinette that is thinner and therefore uses less yarn.

Any yarn can be either knitted or crocheted, provided needles or hooks of the correct size are used, but the cord’s properties should be taken into account. For example, lofty, thick woolen yarns tend to function better when knitted, which does not crush their airy structure, while thin and tightly spun yarn helps to achieve the firm texture required for Amigurumi crochet.^[58]

It has been very common for people and groups to crochet clothing and other garments and then donate them to soldiers during war. People have also crocheted clothing and then donated it to hospitals, for sick patients and also for newborn babies. Sometimes groups will crochet for a specific charity purpose, such as crocheting for homeless shelters, nursing homes, etc.

It is becoming increasingly popular to crochet hats (commonly referred to as “chemo caps”) and donate them to cancer treatment centers, for those undergoing chemotherapy and therefore losing hair. During October pink hats and scarves are made and proceeds are donated to breast cancer funds. Organizations dedicated to using crochet as a way to help others include Knots of Love, Crochet for Cancer,^[59] and Soldiers’ Angels.^[60] These organizations offer warm useful items for people in need.

In 2020, people around the world banded together to help save the wildlife affected by the Australian bushfires by crocheting kangaroo pouches, koala mittens, and wildlife nests.^[61] This was an international effort to help during the particularly bad bushfire season which devastated local ecological systems.

A group started in 2005 to create crochet versions of coral reefs grew by 2022 to over 20,000 contributors in what became the Crochet Coral Reef Project.^[62] To promote awareness of the effects of global warming, their creations have been displayed in galleries and museums, and have been seen by an estimated 2 million people.^[62] Many creations apply hyperbolic (curved) geometric shapes—distinguished from Euclidean (flat) geometry—to emulate natural structures.^[62] Extending hyperbolic crochet for activism and education with color, a group of South African crafters created The Abundance Crochet Coral Reef, an eco-art installation in Cape Town’s Two Oceans Aquarium, to juxtapose hyperbolic shapes crocheted in variations of white on one side of a display with fiber coral shapes crocheted in various colors to illustrate coral bleaching due to oceanic warming and climate change.^[63]

Feminist scholar-activists have argued for crochet as an embodied method of inquiry aimed at uncovering entangled, relational, and situated ways being and knowing inclusive of the more-than-human co-creation of worlds.^[64] In Staying with the Trouble, Donna Haraway argues for the methodological use of crochet to model ecological and mathematical phenomena as “a kind of lure to an affective cognitive ecology stitched in fiber arts” that works “not by mimicry, but by open-ended, exploratory process.”^[40]

In recent years, a practice called yarn bombing, or the use of knitted or crocheted cloth to modify and beautify one’s (usually outdoor) surroundings, emerged in the US and spread worldwide.^[65] Yarn bombers sometimes target existing pieces of graffiti for beautification. In 2010, an entity dubbed “the Midnight Knitter” hit West Cape May. Residents awoke to find knit cozies hugging tree branches and sign poles.^[66] In September 2015, Grace Brett was named “The World’s Oldest Yarn Bomber”. She is part of a group of yarn graffiti-artists called the Souter Stormers, who beautify their local town in Scotland.^[67] In 2006, protesters in Denmark, the United Kingdom, and the United States yarnbombed a World War II combat tank with four thousand pink knitted squares.^[68]

Crochet has been used to illustrate shapes in hyperbolic space that are difficult to reproduce using other media or are difficult to understand when viewed two-dimensionally.^[69] Mathematician Daina Taimiņa first used crochet in 1997 to create strong, durable models of hyperbolic space after finding paper models were delicate and hard to create. These models enable one to turn, fold, and otherwise manipulate space to more fully grasp ideas such as how a line can appear curved in hyperbolic space yet actually be straight. Her work received an exhibition by the Institute For Figuring.^[69]

Examples in nature of organisms that show hyperbolic structures include lettuces, sea slugs, flatworms and coral. Margaret Wertheim and Christine Wertheim of the Institute For Figuring created a traveling art installation of a coral reef using Taimina’s method. Local artists are encouraged to create their own “satellite reefs” to be included alongside the original display.^[70] As hyperbolic and mathematics-based crochet has become more popular, there have been several events highlighting work from various fiber artists. Two shows were Sant Ocean Hall at the Smithsonian in Washington, D.C., and Sticks, Hooks, and the Mobius: Knit and Crochet Go Cerebral at Lafayette College in Pennsylvania.^[71]

In Style in the technical arts, Gottfried Semper looks at the textile with great promise and historical precedent. In Section 53, he writes of the “loop stitch, or Noeud Coulant: a knot that, if untied, causes the whole system to unravel.” In the same section, Semper confesses his ignorance of the subject of crochet but believes strongly that it is a technique of great value as a textile technique and possibly something more

There are a small number of architects currently interested in the subject of crochet as it relates to architecture. The following publications, explorations and thesis projects can be used as a resource to see how crochet is being used within the capacity of architecture.

• Emergent Explorations: Analog and Digital Scripting – Alexander Worden
• Research and Design: The Architecture of Variation – Lars Spuybroek
• YurtAlert – Kate Pokorny

• Mosaic crochet
• Granny square
• Freeform crochet
• Motifs
• Crocheted lace
• Tunisian crochet
• Tapestry crochet
• Amigurumi
• Filet crochet
• Corner to Corner (C2C) Crochet
• Irish crochet lace
• Bead crochet
• Doily

• List of crochet stitches
• Crochet Guild of America
• The Tempestry Project
• Fiber art
• Macramé
• Tatting
• Knitting
• Sewing
• Embroidery

57. York, Portia et al. “Crochet: Engaging Secondary School Girls in Art for STEAM’s Sake.” Science Education International 33.4 (2022): 392–399. Science Education International. Web.

• The Antique Pattern Library

Square knot insignia are embroidered cloth patches that represent awards of the Scout associations throughout the world.

The Scout Association of the United Kingdom uses a “figure-eight” knot and many Scouting organizations of the Commonwealth countries follow suit. The World Organization of the Scout Movement uses military-style ribbons. Scouting America uses a square knot made of colored ropes is depicted; the colors are generally dictated by the award the insignia is associated with.^[1][2]

In the earliest days of the Scouting Movement military veterans were urged into service as Scoutmasters. The first Scout uniforms therefore resembled military uniforms. It was common for these veterans to wear their military decorations on their modified Boy Scout uniform — a national uniform was not to be developed until the early 1920s.^[2]

Military tradition dictated that the actual medal from a military award was only worn on ceremonial occasions — at other times, it was replaced with a thin ribbon bar with the same ribbon style as found attached to the medal. This carried over to Scouting, whose awards were medals, similar to the military, but were most often worn as ribbons.

The first country to switch over from military ribbons to a unique parallel was the United Kingdom, which introduced its knot emblems in 1922.^[3]

The Boy Scouts of America likewise moved away from allowing Scouters to wear military ribbons, but kept the style, introducing their own ribbons in place of medals in 1934. The BSA introduced its own square knot insignia in lieu of the military-style ribbons in 1947. The choice of the square knot as the common emblem was made by James E. West, who is said to have chosen it for its use as the knot associated with first aid, thereby reminding Scouts to continue to be of service to others.^[2]

The first eight awards with square knot insignia in the BSA were the Eagle Scout Award, Quartermaster Award, Scouter’s Training Award, Scouter’s Key, Silver Beaver Award, Silver Antelope Award, Silver Buffalo Award, and Honor Medal.

Since the introduction of square knot insignia, over forty new awards have been added, then combined, and some retired. Currently, there are 32 nationally authorized square knot insignia representing various medals, medallions, certificates, plaques, and other awards.

There are a few cases of local Council-approved square knots and various popular unofficial or spoof knots.

In Scouting America, square knot insignia are worn in rows of three across the top of the wearer’s left pocket. Though there is no rule limiting the number of such insignia that may be worn, some suggest limiting to three rows of three. When fewer than three knots are worn in a particular row, knots are generally centered to the pocket, or the row below.^[4]

Dates indicated in the chart below indicate the use of the square knot insignia for the particular award, and may not correspond to the history of the award itself. For example, the first Eagle Scout Award was presented in 1912, and there was a ribbon bar for it from 1934. It was among the first eight awards given square knot insignia in 1947, so that later date is indicated.

Achievement Awards

These awards are generally earned by the Scouts themselves.
These represent the highest youth rank or achievement in each respective program of Scouting America.

Insignia	Title	Started	Description
	Arrow of Light (Cub Scouts)	1979	Highest rank of the Cub Scouting program (ages 5-11)
	Eagle Scout (Scouts BSA)	1947	Highest rank of the Scouting program (ages 11-18)
	Eagle Scout with Life Membership in NESA	2008	Indicates lifetime membership in the National Eagle Scout Association
	Quartermaster Award (Sea Scouts)	1947	Highest rank in the Sea Scouting program (ages 14-20)
	Silver Award / Summit Award (Venturing)	1999	Highest rank in the Venturing program (ages 14-20)

Distinguished service

These awards are generally granted via nomination and review; they cannot be earned or applied for by Scouts or Scouters directly.

Insignia	Title	Started	Description
	Unit Leader Award of Merit	2010	Unit-level quality leader recognition award
	District Award of Merit	1971	District-level distinguished service award
	Silver Beaver Award	1947	Council-level distinguished service award
	Silver Antelope Award	1947	Territory-level distinguished service award
	Silver Buffalo Award	1947	National-level distinguished service award
	Silver World Award	1976	International-level distinguished service award

Heroism

These awards are generally granted via nomination and review; they cannot be earned or applied for by Scouts or Scouters directly.

Insignia	Title	Started	Description
	Medal of Merit	1952	Performed an act of service of a rare or exceptional character that reflects an uncommon degree of concern for the well-being of others.
	Heroism Award	1983	Demonstrated heroism and skill in averting serious injury or saving or attempting to save life at minimum risk to self. Discontinued late 2012; reinstated February 2018
	Honor Medal	1947	Demonstrated unusual heroism and skill or resourcefulness in saving or attempting to save life at considerable risk to self. Awarded with crossed palms for situations with extreme risk to self.

Leadership and training

These awards are generally earned by the Scouters themselves.

Insignia	Title	Started	Description
	Den Leader Training Award	1989
	Scouter’s Training Award	1947
	Scouter’s Key	1947
	Commissioner Award of Excellence in Unit Service	2011
	Doctorate of Commissioner Science	2008
	Philmont Training Center Masters Track Award	2008
	Professional Training Award	1980

Special achievements

These awards are generally earned by the Scouts and Scouters themselves.
In one case (James E. West Fellowships) nomination by others is also possible.

Insignia	Title	Started	Description
	Youth Religious Emblems[5]	1971
	Distinguished Conservation Service Award	2020	Replaced William T. Hornaday Awards
	James E. West Fellowship	1993
	International Scouter Award	2002
	William D. Boyce New-Unit Organizer Award	2005
	Alumni Award	2011	Presented to Scouting alumni who promote continued engagement with Scouting America

Special service recognition

These awards are generally granted via nomination and review; they cannot be earned or applied for by Scouts or Scouters directly.

Insignia	Title	Started	Description
	Adult Religious Awards[6]	1973	Distinguished Service to Scouts and Youth associated with a religious community or organization
	Order of the Arrow Distinguished Service Award	1976	Distinguished service to the Order of the Arrow
	Distinguished Commissioner Service Award	1987	Distinguished Service as a Commissioner at the unit, district, or council level
	Community Organization Awards[7]	2002	Distinguished Service to Scouts and Youth affiliated with a non-religious community organization
	Venturing Leadership Award	2004	Distinguished Service to Venturing and Venturers
	Scouting Service Award [8]	2017	Distinguished Service to Scouts of a specific ethnic or special needs identity; Incorporates – Whitney M. Young Jr. Service Award, Scouting … Vale la pena! Service Award, Asian American Spirit of Scouting Service Award, American Indian Scouting Association Grey Wolf Award, Special Needs Scouting Service Award

Discontinued

These awards have been completely discontinued or combined and represented by current award knots.

Insignia	Title	Started	Ended	Description
	William T. Hornaday Medals	1991	2020	Superseded by Distinguished Conservation Service Award
	Skipper’s Key	1947	1949	Superseded by Scouter’s Key
	Exploring Ranger Award	1933	1951	Superseded by “Ranger” bar
	Air Scout Ace Award	1950	1954
	Exploring Silver Award	1954	1958
	Exploring Awards Knot[10]	1954	1998
	William H. Spurgeon III Award[11]	1989	2007	Recognition for individuals and organizations contributing significant leadership to the Exploring program[11]
	Scoutmaster Award of Merit	1987	2010	Superseded by Unit Leader Award of Merit
	Venturing Advisor / Varsity Coach Award of Merit	1999	2010	Superseded by Unit Leader Award of Merit
	Pack Trainer Award	2006	2011	Superseded by Scouter’s Training Award[12]
	Cubmaster Award	1989	2012	Superseded by Scouter’s Key
	Cub Scouter Award	1989	2012	Superseded by Scouter’s Training Award
	Den Leader Coach Award	1989	2012	Superseded by Den Leader Training Award
	Webelos Den Leader Award	1989	2012	Superseded by Den Leader Training Award
	Tiger Cub Den Leader Award	1992	2012	Superseded by Den Leader Training Award
	George Meany Award	1987	2011	Incorporated in Community Organization Award
	Sea Badge	1989	2012	Superseded by Seabadge Trident pin
	Asian American Spirit of Scouting Service Award	2003	2016	Incorporated in Scouting Service Award
	¡Scouting…Vale la Pena! Service Award	2003	2016	Incorporated in Scouting Service Award
	Whitney M. Young Jr. Service Award	1989	2016	Incorporated in Scouting Service Award
	Speakers Bank Award	2007	2011	Discontinued

The British Scout Association introduced the idea of a cloth ‘knot’ insignia in 1922, modeled after the military use of ribbons to represent medals and other citations. Many of the Scouting organizations of the Commonwealth of Nations either copy the British system or simply use the British awards and insignia.

Gallantry

Insignia	Title	Started	Description
	Bronze Cross	Year	The Bronze Cross is the highest gallantry award given by the Scout Association UK for special heroism or action in the face of extraordinary risk.
	Silver Cross	Year	The Silver Cross Award is a prestigious accolade within the UK Scout Association, awarded for gallantry in circumstances of considerable risk. This award is a testament to the recipient’s bravery and commitment to duty, often in challenging and perilous situations.
	Gilt Cross	Year	The Gilt Cross Award is a prestigious accolade within the UK Scout system, awarded for gallantry in circumstances of moderate risk. It is a symbol of bravery and acts of heroism where life is at risk.
	Cornwell Scout Badge	Year	is a gallantry award of The Scout Association of the United Kingdom and some of its branches in other countries. It is awarded in recognition of devotion to duty, courage and endurance.

Meritorious Conduct

Insignia	Title	Started	Description
	Medal for Meritorious Conduct	Year	The Chief Scouts Medal for Meritorious Conduct is the highest award for acts of bravery and service within the UK Scouts.
	Chief Scout’s Commendation for Meritorious Conduct	Year	The Chief Scout’s Commendation for Meritorious Conduct is one of the highest awards within the UK Scouts. It is awarded for acts of bravery, dedication, and outstanding service to the Scouts.

Service

Insignia	Title	Started	Description
	Silver Wolf	Year	The Silver Wolf is the highest award made by The Scout Association (UK) “for services of the most exceptional character.” It is an unrestricted gift of the Chief Scout.
	Silver Acorn	Year	The Silver Acorn is awarded after at least 20 years’ service, which should be specially distinguished and appreciably better than outstanding.
	Award for Merit	Year	The Award of Merit is awarded for outstanding service. It implies keen, conscientious, imaginative and dedicated service over a sustained period, of at least 10 years duration.
	Chief Scout’s Commendation for Good Service	Year	The Chief Scout’s Commendation is awarded in respect of not less than 5 years good service, which stands out. It should be regarded as the Chief Scout’s recognition of the very real contribution made to the Scouts by the individual concerned.
	Chief Scout’s Personal Award	Year	The Chief Scout’s Award is a prestigious recognition in the UK Scouting system, awarded to individuals for their outstanding contributions to Scouting. The award is available in various levels, each representing a different age group and level of achievement.
	Commissioner’s Commendation Award	Year	The Commissioners Commendation Award in the UK is a local award used to recognize individuals, including youth members, adult members, and non-members, for their contributions to Scouting.
	Length of Service Decoration	Year	The Chief Scout’s Length of Service Awards in the UK are presented to recognize the significant contributions of adult members to the Scout Association. These awards are given at intervals of 5, 10, 15, 20, 25, 30, 40, 50, 60, and 70 years of service.

• “Awards Central”. Scouting.org. Boy Scouts of America.
• “Awards-Knots”. U.S. Scouting Service Project, Inc. Retrieved June 10, 2013.
• Crowl, George. “Illustrated History of BSA Square Knot Evolution”. BSA Badge History.
• Crowl, George. “Knots, Pins, and Insignia of the Boy Scouts of America”. Crowl Family Webpages. Retrieved March 27, 2022.
• Kauffmann, Michael. “Scouter Knots”. MeritBadge.org. Archived from the original on March 10, 2019.
• Walton, Mike. “Knot Emblems of the BSA”. Badge and Uniform Site.

• Whitman, Gary (2014). Ribbon Bars & Knot Awards of the Boy Scouts of America (7th ed.). CreateSpace. ISBN 978-1499201277.

The Croatian interlace or Croatian wattle, known as the pleter or troplet in Croatian, is a type of interlace, most characteristic for its three-ribbon pattern. It is one of the most often used patterns of Croatian pre-Romanesque art and architecture.^[1]

It is found on and within churches as well as monasteries built in early medieval Duchy of Croatia and Kingdom of Croatia between the 9th and beginning of the 12th century, with examples counting in “hundreds”, making Croatia country with highest concentration of interlace in general in Europe.^[2]

The interlaces in Croatia are dominated by double and triple ribbons.^[2] The ornamental strings were sometimes grouped together with animal and herbal figures. Most representative examples of inscriptions embellished with the interlace include the Baška tablet, Baptismal font of Prince Višeslav and the Branimir Inscription. Other notable examples are located near Knin, in Ždrapanj and Žavić by the Bribir settlement, Rižinice near Solin and in Split and Zadar.

There is a scholarly debate about the origin and identity of the regional three-ribbon interlace, some considering it as Croatian “national expression”, other arguing Lombardic influence (introduced by the Patriarchate of Aquileia), but such exclusive models are problematic.^[2][3]

Croatia has a civil and military decoration called the Order of the Croatian Interlace.^[4]

• 
  Inscription of duke Branimir, 879–892
• 
  Pluteus from the 9th century^[3]
• 
  Pluteus from the 9th century^[1]
• 
  Inscription of king Stephen Držislav, mid-10th century
• 
  Part of pluteus from the 11th century^[3]
• 
  Font from the late-11th century (most likely a depiction of Petar Krešimir IV or Demetrius Zvonimir)
• 
  Baška tablet
• 
  Emblem of the Ustaše, a former fascist and ultranationalist organization, as displayed on the state symbols of the Independent State of Croatia
• 
  Croatian interlace bordering the Seal of Armed Chief of General Staff of the Armed Forces of Croatia
• 
  Croatian interlace on the Cross at the confluence of the Vuka and Danube in Vukovar

• National symbols of Croatia
• Interlace (art)

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.