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  • Milnor map

    In mathematics, Milnor maps are named in honor of John Milnor, who introduced them to topology and algebraic geometry in his book Singular Points of Complex Hypersurfaces (Princeton University Press, 1968) and earlier lectures. The most studied Milnor maps are actually fibrations, and the phrase Milnor fibration is more commonly encountered in the mathematical literature. These were introduced to study isolated singularities by constructing numerical invariants related to the topology of a smooth deformation of the singular space.

    Definition

    Let f ( z 0 , … , z n ) {\displaystyle f(z_{0},\dots ,z_{n})} {\displaystyle f(z_{0},\dots ,z_{n})} be a non-constant polynomial function of n + 1 {\displaystyle n+1} {\displaystyle n+1} complex variables z 0 , … , z n {\displaystyle z_{0},\dots ,z_{n}} {\displaystyle z_{0},\dots ,z_{n}} where the vanishing locus of

    f ( z )    and    ∂ f ∂ z i ( z ) {\displaystyle f(z)\ {\text{ and }}\ {\frac {\partial f}{\partial z_{i}}}(z)} {\displaystyle f(z)\ {\text{ and }}\ {\frac {\partial f}{\partial z_{i}}}(z)}

    is only at the origin, meaning the associated variety X = V ( f ) {\displaystyle X=V(f)} {\displaystyle X=V(f)} is not smooth at the origin. Then, for K = X ∩ S ε 2 n + 1 {\displaystyle K=X\cap S_{\varepsilon }^{2n+1}} {\displaystyle K=X\cap S_{\varepsilon }^{2n+1}} (a sphere inside C n + 1 {\displaystyle \mathbb {C} ^{n+1}} {\displaystyle \mathbb {C} ^{n+1}} of radius ε > 0 {\displaystyle \varepsilon >0} {\displaystyle \varepsilon >0}) the Milnor fibration[1]pg 68 associated to f {\displaystyle f} {\displaystyle f} is defined as the map

    ϕ : ( S ε 2 n + 1 ∖ K ) → S 1    sending    x ↦ f ( x ) | f ( x ) | {\displaystyle \phi \colon (S_{\varepsilon }^{2n+1}\setminus K)\to S^{1}\ {\text{ sending }}\ x\mapsto {\frac {f(x)}{|f(x)|}}} {\displaystyle \phi \colon (S_{\varepsilon }^{2n+1}\setminus K)\to S^{1}\ {\text{ sending }}\ x\mapsto {\frac {f(x)}{|f(x)|}}},

    which is a locally trivial smooth fibration for sufficiently small ε {\displaystyle \varepsilon } {\displaystyle \varepsilon }. Originally this was proven as a theorem by Milnor, but was later taken as the definition of a Milnor fibration. Note this is a well defined map since

    f ( x ) = | f ( x ) | ⋅ e 2 π i Arg ⁡ ( f ( x ) ) {\displaystyle f(x)=|f(x)|\cdot e^{2\pi i\operatorname {Arg} (f(x))}} {\displaystyle f(x)=|f(x)|\cdot e^{2\pi i\operatorname {Arg} (f(x))}},

    where Arg ⁡ ( f ( x ) ) {\displaystyle \operatorname {Arg} (f(x))} {\displaystyle \operatorname {Arg} (f(x))} is the argument of a complex number.

    Historical motivation

    One of the original motivations for studying such maps was in the study of knots constructed by taking an ε {\displaystyle \varepsilon } {\displaystyle \varepsilon }-ball around a singular point of a plane curve, which is isomorphic to a real 4-dimensional ball, and looking at the knot inside the boundary, which is a 1-manifold inside of a 3-sphere. Since this concept could be generalized to hypersurfaces with isolated singularities, Milnor introduced the subject and proved his theorem.

    In algebraic geometry

    Another closed related notion in algebraic geometry is the Milnor fiber of an isolated hypersurface singularity. This has a similar setup, where a polynomial f {\displaystyle f} {\displaystyle f} with f = 0 {\displaystyle f=0} {\displaystyle f=0} having a singularity at the origin, but now the polynomial

    f t : C n + 1 → C    sending    ( z 0 , … , z n ) ↦ f ( z 0 , … , z n ) − t {\displaystyle f_{t}\colon \mathbb {C} ^{n+1}\to \mathbb {C} \ {\text{ sending }}\ (z_{0},\ldots ,z_{n})\mapsto f(z_{0},\ldots ,z_{n})-t} {\displaystyle f_{t}\colon \mathbb {C} ^{n+1}\to \mathbb {C} \ {\text{ sending }}\ (z_{0},\ldots ,z_{n})\mapsto f(z_{0},\ldots ,z_{n})-t}

    is considered. Then, the algebraic Milnor fiber is taken as one of the polynomials f t ≠ 0 {\displaystyle f_{t\neq 0}} {\displaystyle f_{t\neq 0}}.

    Properties and Theorems

    Parallelizability

    One of the basic structure theorems about Milnor fibers is they are parallelizable manifolds[1]pg 75.

    Homotopy type

    Milnor fibers are special because they have the homotopy type of a bouquet of spheres[1]pg 78. The number of these spheres is the Milnor number. In fact, the number of spheres can be computed using the formula

    μ ( f ) = dim C C [ [ z 0 , … , z n ] ] Jac ⁡ ( f ) , {\displaystyle \mu (f)={\text{dim}}_{\mathbb {C} }{\frac {\mathbb {C} [[z_{0},\ldots ,z_{n}]]}{\operatorname {Jac} (f)}},} {\displaystyle \mu (f)={\text{dim}}_{\mathbb {C} }{\frac {\mathbb {C} [[z_{0},\ldots ,z_{n}]]}{\operatorname {Jac} (f)}},}

    where the quotient ideal is the Jacobian ideal, defined by the partial derivatives ∂ f / ∂ z i {\displaystyle \partial f/\partial z_{i}} {\displaystyle \partial f/\partial z_{i}}. These spheres deformed to the algebraic Milnor fiber are the Vanishing cycles of the fibration[1]pg 83. Computing the eigenvalues of their monodromy is computationally challenging and requires advanced techniques such as b-functions[2]pg 23.

    Milnor’s fibration theorem

    Milnor’s Fibration Theorem states that, for every f {\displaystyle f} {\displaystyle f} such that the origin is a singular point of the hypersurface V f {\displaystyle V_{f}} {\displaystyle V_{f}} (in particular, for every non-constant square-free polynomial f {\displaystyle f} {\displaystyle f} of two variables, the case of plane curves), then for ϵ {\displaystyle \epsilon } {\displaystyle \epsilon } sufficiently small,

    f | f | : ( S ε 2 n + 1 ∖ V f ) → S 1 {\displaystyle {\dfrac {f}{|f|}}\colon \left(S_{\varepsilon }^{2n+1}\setminus V_{f}\right)\to S^{1}} {\displaystyle {\dfrac {f}{|f|}}\colon \left(S_{\varepsilon }^{2n+1}\setminus V_{f}\right)\to S^{1}}

    is a fibration. Each fiber is a non-compact differentiable manifold of real dimension 2 n {\displaystyle 2n} {\displaystyle 2n}. Note that the closure of each fiber is a compact manifold with boundary. Here the boundary corresponds to the intersection of V f {\displaystyle V_{f}} {\displaystyle V_{f}} with the ( 2 n + 1 ) {\displaystyle (2n+1)} {\displaystyle (2n+1)}-sphere (of sufficiently small radius) and therefore it is a real manifold of dimension ( 2 n − 1 ) {\displaystyle (2n-1)} {\displaystyle (2n-1)}. Furthermore, this compact manifold with boundary, which is known as the Milnor fiber (of the isolated singular point of V f {\displaystyle V_{f}} {\displaystyle V_{f}} at the origin), is diffeomorphic to the intersection of the closed ( 2 n + 2 ) {\displaystyle (2n+2)} {\displaystyle (2n+2)}-ball (bounded by the small ( 2 n + 1 ) {\displaystyle (2n+1)} {\displaystyle (2n+1)}-sphere) with the (non-singular) hypersurface V g {\displaystyle V_{g}} {\displaystyle V_{g}} where g = f − e {\displaystyle g=f-e} {\displaystyle g=f-e} and e {\displaystyle e} {\displaystyle e} is any sufficiently small non-zero complex number. This small piece of hypersurface is also called a Milnor fiber.

    Milnor maps at other radii are not always fibrations, but they still have many interesting properties. For most (but not all) polynomials, the Milnor map at infinity (that is, at any sufficiently large radius) is again a fibration.

    Examples

    The Milnor map of f ( z , w ) = z 2 + w 3 {\displaystyle f(z,w)=z^{2}+w^{3}} {\displaystyle f(z,w)=z^{2}+w^{3}} at any radius is a fibration; this construction gives the trefoil knot its structure as a fibered knot.

    See also

    • Vanishing cycle
    • Mixed Hodge structure

    References

    1. 1 2 3 4 Dimca, Alexandru (1992). Singularities and Topology of Hypersurfaces. New York, NY: Springer. ISBN 978-1-4612-4404-2. OCLC 852790417.
    2. ↑ Budur, Nero. “Multiplier ideals, Milnor fibers, and other singularity invariants” (PDF). doi:10.1002/humu.22655. S2CID 221776902. Archived from the original (PDF) on 6 March 2019.
    • Milnor, John W. (1968), Singular points of complex hypersurfaces, Annals of Mathematics Studies, No. 61. Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, ISBN 0-691-08065-8


    This article is adapted from “Milnor map” on Wikipedia, written by its contributors, and used under CC BY-SA 4.0. Text on this page is available under the same licence. Source snapshot: Wikipedia via Kiwix, 2026-07-20.

  • Aiguillette

    Aiguillette
    Commissioned officers (front row) and non-commissioned officers (second row) of the Household Cavalry in full dress wearing aiguillettes
    Aiguillette
    Bundle of 20 braided gold and silver laces with stamped brass tags or aiglets, first half of the 17th century

    An aiguillette (French: [ɛɡɥijɛt] ⓘ, from aiguille, “needle”), also spelled aguillette, aiglet or aglet, is a cord with metal tips or lace tags, or the decorative tip itself.[1]: 4 [2]

    Functional or purely decorative fasteners of silk cord with metal tips were popular in the 16th and early 17th centuries, sometimes of gold set with gemstones or enameled, are generally called “aiglets”, “aglets” or “points”.[2][3]: 97 

    In modern usage, an “aiguillette” or “axelbant”[4] is an ornamental braided cord with decorative metal tips worn on uniforms or as part of other costumes such as academic dress,[1] where it will denote an honour. This usage of “aiguillette” derives from lacing used to fasten plate armor together. As such, a knot or loop arrangement was used which sometimes hung from the shoulder.

    These aiguillettes should not be confused with lanyards, which are cords also worn from the shoulder (or around the neck), but do not have the pointed aiguillette tips and are usually of fibre rather than gold or silver wire, and often not braided.

    The modern aglet or shoelace tip and the decorative tips on bolo ties are types of aiguillettes.

    History

    Two crystal aiglets
    Ribbon tie with aiglets
    Pair of crystal, gold, and enamel aiglets, late 16th or early 17th century

    The modern aiguillette derives from the laces used to secure plates of armor together. The breast- and back-plates would be attached on one side with short loops of cord acting as a hinge, and on the other by a longer and more ornate tied one, to support the arm defences.[5][6][7]

    Another tradition traces a symbolic use of a Aiguillettes to Spain, after the Duke of Alba condemned deserting Flemish to face the rope. The Flemish decided to wear a cord and a nail on their collar as a symbol of their defiance. The cord and the nail came seen as an emblem of their bravery and ultimately became the honour known as aiguillettes.[8][9]

    Portraits of the 16th and 17th centuries show that aiglets or metal tips could be functional or purely decorative, though many were used to “close” seams and slashes that are not always apparent on dark garments in portraits. They were made in matched sets, might be of silver, silver-gilt, or gold, and were worn in masses.[3]

    A 1547 inventory of Henry VIII of England’s wardrobe includes one coat with 12 pairs of aiglets, and 11 gowns with a total of 367 pairs.[2] The Day Book of the Wardrobe of Robes of Elizabeth I records items received into storage, including details of buttons and aiglets lost from the Queen’s clothing. This entry suggests the huge numbers of matching aiglets fashionable forty years later:

    Lost the 2 of February … [1582] 1 bunsh of small gold tagges or aglettes from a gowne of black satten at Sittingbourne parcell [part] of uppon the same gowne 193 bunshes[10]: 74 

    Elizabeth’s aiglets were variously enameled with white, red, black, blue, and purple details or set with diamonds, garnets, rubies and pearls.[3][10] In Scotland, they were known as “horns”, Mary, Queen of Scots had pairs of gold horns enamelled in red and white and set with pearls.[11] Those of Anne of Denmark in the early years of the 17th century were larger, shaped in triangles and pyramids. One set of 24 were made three-sided, with “27 diamonds in the sides and one in the top”, for a total of 642 diamonds in the set.[3]

    Gallery

    • Surviving 16th or 17th-century copper-alloy lace tag found in London.
      Surviving 16th or 17th-century copper-alloy lace tag found in London.
    • Hat and gown trimmed with pairs of aiglets, 1538
      Hat and gown trimmed with pairs of aiglets, 1538
    • Gowns decorated with gold aiglets, 1559
      Gowns decorated with gold aiglets, 1559
    • Rose ribbons tipped with large aiglets set with pearls, c. 1560s
      Rose ribbons tipped with large aiglets set with pearls, c. 1560s
    • Doublet and hose with ribbon points tipped with aiglets, 1630s
      Doublet and hose with ribbon points tipped with aiglets, 1630s
    • Surviving lace tag, ca. 1500–1700, found in Lincolnshire
      Surviving lace tag, ca. 1500–1700, found in Lincolnshire

    Military usage

    Aiguillette
    Lord High Admiral Clinton in an arming doublet of 1562, showing the decorative points or laces with metal tips for securing armour.

    Argentina

    Aiguillettes are worn on the right shoulder by armed forces officers serving in specific positions, such as aide-de-camp to the President, the Minister of Defense, each service’s chief of staff and unit commanding officers, and by military attachés to Argentinian embassies abroad. The color of the aiguillette may be golden, silver or tan, depending the nature of the assignment.

    Olive green aiguillettes are worn with the combat uniform in very special circumstances, such as ceremonies and inspections. Otherwise, aiguillettes are not worn with the combat uniform. A special red aiguillette is worn by the adjutants to the commanding officers of the Horse Grenadiers Regiment (the presidential guard) and the Military Academy. Also, a red aiguillette is worn on the left shoulder by the senior NCO of each Army unit. A thinner, yellow aiguillette is worn on the right shoulder by NCOs who have completed the instructor course.

    In the Navy, adjutants to very senior officers wear golden aiguillettes on the left shoulder.

    Australia

    Aiguillettes distinguish officers of Flag, General and Air rank in specific command appointments, military attachés and aides-de-camp. Most senior officers and aides-de-camp to the Governor-General or state governors wear the aiguillette on the right shoulder, whilst military attachés and staff aides-de-camp wear the aiguillette on the left. Royal aiguillettes are of plain gold, naval aiguillettes are of blue and gold, army aiguillettes are of red and gold, air force aiguillettes of light blue and gold.

    The Governor-General of Australia, as the Commander in Chief of the Australian Defence Force, is entitled to wear a uniform on which an aiguillette made of platinum is worn.[12]

    Canada

    Aiguillettes are worn by officers in specific appointments to specific principals. For the following appointments the aiguillette is worn on the right shoulder:

    • Aide-de-camp to His Majesty the King of Canada and other members of the Royal family;
    • Aide-de-camp to the Governor-General of Canada;
    • Aide-de-camp to Lieutenant-Governors of provinces;
    • Aide-de-camp to visiting foreign heads of state; and
    • Honorary member of the Royal Medical Household.

    For other appointments, the aiguillette is worn on the left shoulder:

    • Attachés as part of an embassy or consulate;
    • Staff officer to the Minister of National Defence;
    • Aides-de-camp / flag lieutenants to general / flag officers; and
    • Officers appointed as personal assistants to senior officers, Ministers of the Crown, visiting military or civilian officials, dignitaries of foreign nations.

    For all regal and vice-regal appointments, the distinguishing badge or cipher of the principal is worn on the shoulder strap or shoulder board.

    Obsolete pattern aiguillettes braided with a coloured stripe may be worn until replaced by officers holding appointments to Lieutenant-Governors of provinces or military / civilian principals.[13]

    For RCN officers, since the uniform jacket does not normally include shoulder straps, the aiguillette is secured under a fully-corded shoulderboard worn on the right shoulder for regal and vice-regal appointments, and under a hard shoulderboard with a corded knot worn on the left shoulder for all other appointments.

    For RCAF officers, since the uniform jacket does not normally include shoulder straps, there is a small attachment hook worn on the appropriate shoulder to which the aiguillette is fastened.

    For Army officers, the aiguillette is secured under the appropriate shoulder strap (left or right) and the cipher or badge (if appropriate) is worn on that shoulder strap.

    For all branches, the end of the cord near the hanging ends is fastened to a concealed button under the left or right jacket lapel as appropriate.[14]

    Denmark

    Aiguillette
    Royal Danish Naval Band with red aiguillettes

    The Danish military uses aiguillettes for a number of different positions.

    • Blue and yellow: Danish troops stationed with UNFICYP
    • Blue and red: the Bugle Corps of the Guard Hussar Regiment Mounted Squadron
    • Red: Royal Danish Naval Band
    • Green: Danish military police
    • Gold: Adjutants

    France

    Aiguillette
    Republican Guard wearing a gold-wire aiguillette on the left shoulder

    The aiguillettes are worn only with the dress uniform. There are several types of aiguillettes in the French military:

    • Aiguillettes on the right shoulder are worn by officers filling certain duties.
      • Staff officers in the President’s staff, the Prime Minister’s staff, the Minister of Defence’s staff and the Maréchaux’s Staff. The aiguillette is always in gold wire.
      • The aides-de-camp to the President, the Prime Minister, the Chancellor of the Légion d’honneur, the Chief of the Defence Staff and the DGA. The aiguillette can be silver or gold, depending on the officer’s corps. The AdC to the generals in the army and the air force and all the staff officers in the navy can also wear this aiguillette.
      • The military attachés of an embassy. The aiguillette is always in gold wire.
    • Aiguillettes on the left shoulder are worn by all members of some units:
      • The Gendarmes. The aiguillette is white for the Gendarmerie départementale and gold for the Gendarmerie mobile, including the Garde Républicaine.
      • The Bands. This aiguillette can be blue, red, or gold.

    Aiguillettes should not be confused with fourragères, which can be worn in the same way on the left shoulder in parade dress.

    Germany

    In the German armies, including the Imperial German Army and the Imperial German Navy, the adjutant generals or admirals wore the Adjutantenschnur or Achselband (adjutant cord or aguilette) on the right shoulder as of 1863 in gold, the generals à la suite and the Flügeladjutanten in silver. There were initially several experimental forms in the Reichswehr.

    On 29 June 1935, so-called shoulder straps were introduced into the German Wehrmacht and were worn on certain occasions, e.g. at parades. Adjutants wore armpit cords as their badge of activity. In the National People’s Army, the border troops of the GDR and the GDR People’s Navy, a silver-colored armpit cord with silver tips for LaSK/LSK/LV/GT and gold-colored tips for the People’s Navy has also been introduced for officers since 1976; the version for generals and admirals was gold-colored. In the Bundeswehr it is worn by attachés, officers of protocol, flag escort officers, greeting and liaison officers of the Navy as well as (if these tasks are carried out by officers) wake guards and medal cushion bearers.

    Ghana

    In Ghana, aiguillettes form part of the ceremonial uniforms of commissioned officers in the army and other security services including the police, prisons service, fire service, and customs and immigration. Senior police officers wear white aiguillettes with dark-blue uniforms, prison officers also wear yellow aiguillettes over the official ceremonial number one uniform while customs and immigration officials wear red aiguillettes with olive-green outfits. When worn, the aiguillettes denote on-duty status.

    Greece

    Aiguillettes are worn by the aides-de-camp to the President of the Hellenic Republic. The President’s Adjutants wears it on the right shoulder.

    Ireland

    Aiguillettes are worn by the aides-de-camp to the President and Taoiseach. The President’s ADC wears it on the right shoulder, the Taoiseach’s ADC on the left.

    Civil Defence (Cosaint Sibhialta) personnel wear a white aiguillette on their dress uniform.

    Gold aiguillettes are also worn by officers in the Defence Forces with their mess dress uniform. They are worn on the left hand shoulder.

    Israel

    In the IDF, soldiers who wear an aiguillette are mostly instructors.
    List of aiguillette’s colors and roles in the IDF:

    Aiguillette
    IDF uniforms with azure & yellow aiguillette of the casualties & City Officers administration
    • Green & White- Military intelligence instructors (mainly)
    • Black- training staff (mainly)
    • Blue – Navy instructors
    • Blue & Red- Military Police
    • Red (right side)- IDF orchestra
    • Red (Left side)- Senior Navy instructors (mainly)
    • Purple- Service Rights Staff
    • Blue & White- Unit’s NCO
    • White- Security guards
    • Gray- Education instructors
    • Green- General instructors
    • Brown- Diagnostic crew (mainly)
    • Gold- discipline NCO
    • Orange & Azure- Search and Rescue units
    • Orange & Black- Combat Engineering Corps
    • Yellow & Black- WMD instructors
    • Azure- Off-road vehicle operators (mainly)
    • Azure & Green- Off-road vehicle operators instructors
    • Azure & Yellow- Casualties & City Officers administration
    • Azure & Purple- Reserve mobilization
    • Green & Red- Civil population emergency instructors
    • Golden- Military attaché
    • Black & Creme- Military Rabbinate

    In the Israeli Police, policemen mainly wear a black aiguillette, and Israeli Border Police policemen wear a dark green aiguillette.

    Russia

    Aiguillette
    Soldiers of Prezidentskiy polk (The Presidential Regiment), Alexander Garden, Moscow

    Aiguillettes are worn by honor guard personnel. A single silver aiguillette is worn on enlisted cut uniforms of minor detachments. A single gold aiguillette is worn on officer cut uniforms of minor detachments. A double gold aiguillette is worn on officer cut uniforms of major detachments. All personnel of major detachments wear officer cut uniforms. Demobilized soldiers also often decorate their uniforms with makeshift aiguillettes.

    Singapore

    Aiguillettes are worn on the right shoulder by officers of certain appointments only. They include the:

    • Chief of Defence Force (CDF)
    • Commissioner of Police (CP)
    • Commissioner of the Singapore Civil Defence Force
    • Aides-de-camp (ADC) to the President
    • Honorary aides-de-camp to the President
    • Military attachés and assistant military attachés
    • MA/SA/NA to the Chief of Defence Force and service chiefs

    Aiguillettes are also worn on the left shoulder by musicians of the Singapore Armed Forces Band and the Singapore Police Force Band.

    Musicians of the Singapore Armed Forces Band wears a red and gold aiguillette while Director of Musics of the Singapore Armed Forces Band wear a full gold aiguillette.

    Student musicians from both the National Cadet Corps Command Band and National Police Cadet Corps Band similarly wear aiguillettes mirroring the respective parent bodies.

    In Singapore, ADCs who are officers of the Singapore Armed Forces and the Singapore Civil Defence Force wear gold aiguillettes and police officers wear silver aiguillettes. Singapore Armed Forces ADCs wear a gold braid lanyard in lieu of an aiguillette when in No. 3 and No. 5(T).

    Additionally the newly commissioned ADC badges are worn across all five services’ no.4 uniform.

    Sri Lanka

    Aiguillette
    Full Aiguillettes (No. 1A) worn by the Chief of Defence Staff.

    In Sri Lanka, full aiguillettes are worn by members of the personal staff of the President of Sri Lanka and General officers, flag officers and air officers, while half aiguillettes are worn by the Colonels and Lieutenant Colonels in ceremonial uniforms.

    • Aiguillettes (No. 1A) (Full Aiguillettes) of gold cord are worn on the right shoulder by; service commanders and officers of the four-star rank and above on uniforms No. 1, 2, 3, 4.
    • Aiguillettes (No. 1B) (Full Aiguillettes) of gold (and red orris basket in the case of Army) cord are worn on the right shoulder by; Aides-de-camp to the President of Sri Lanka and Aides-de-camp to visiting heads of state (in their presence) on uniforms No. 1.
    • Aiguillettes (No. 2A) (Full Aiguillettes) of gold cord are worn on the left shoulder by; by officers of two-star rank and one-star rank on uniforms No. 1, 2, 3, 4.
    • Aiguillettes (No. 2B) (Full Aiguillettes) of gold (and red orris basket in the case of Army, black basket in the case of Navy) cord are worn on the left shoulder by; Military Assistants and Aides-de-camps to general, flag and air officers.
    • Aiguillettes (No. 3) (Half Aiguillettes) of gold cord joined in the front and back by gilt cloth with a single metal aglet are worn on the left shoulder by Colonels and Lieutenant Colonels in (ceremonial) uniforms No. 1, 2, 3, 4.

    In the Sri Lanka Police, the Inspector General of Police, wear gold aiguillettes on the right shoulder, while Senior gazetted police officers wear black aiguillettes on the left shoulder in both formal and ceremonial dress.

    Sweden

    Aiguillette
    Large aiguillette m/1816, used in Sweden.

    The Swedish Armed Forces uses three types of aiguillettes. All aiguillettes come in gold or silver braid depending on the regiment of the wearer, Royal Guard regiments use silver due to historical reasons, all others gold.

    • Large aiguillette m/1816 (Sw.: Stor ägiljett m/1816) is worn on the right shoulder by officers on the staff of HM the King of Sweden, aide-de-camps of members of the Royal Family and by commissioned colonels and naval captains. It is also permissible to be worn by officers (except generals) of the staff of the Defense Force, and former officers of the Swedish general staff.
    • Small aiguillette m/1889 (Sw.: Liten ägiljett m/1889) is worn on the right shoulder by the on duty adjutant of the King in Service dress uniform, adjutants of government ministers, the Supreme Commander of the Swedish Armed Forces and general officers.
    • Aiguillette m/55 (Sw.: Ägiljett m/55) is worn on the left shoulder by military attachés in Service dress uniform in the country of his or her accreditation; it is only used in Sweden during state visits.

    In Sweden there is a distinction between the aiguillettes mentioned above, and other cords used on uniforms in the Royal Swedish Navy and the Swedish Air Force.

    United Kingdom

    Aiguillette
    Royal Aiguillette worn by the Chief of Defence Staff and Staff Aiguillettes worn by ADCs

    There are four types of aiguillette worn by the British Armed Forces.

    • Aiguillettes (1st Class or Royal) are of gold-wire cord and are worn on the right shoulder by, among others, admirals of the fleet, field marshals and marshals of the RAF; honorary physicians, honorary chaplains, honorary surgeons and aides-de-camp to the Sovereign; and equerries to members of the royal family. Some appointments carry the privilege of wearing a miniature Sovereign’s Cypher on the points of the aiguillettes. These aiguillettes are also worn by commissioned officers of the Household Cavalry (in full dress only). They are worn on the left shoulder in full dress by warrant officers of the Household Cavalry.
    • Aiguillettes (2nd Class or Board) are of gold and dark blue, crimson or light blue depending if worn by Royal Navy, Army or RAF officers and are worn on the right shoulder by, among others, military members of the Defence Board and each Service Board and the personal staff of governors. A simplified version with no coils is worn on the left shoulder by staff corporals, corporals of horse and lance corporals of horse of the Household Cavalry in full dress.
    • Aiguillettes (3rd Class or Staff) are of gold and dark blue, crimson or light blue depending if worn by Royal Navy, Army or RAF officers. They are worn on the left shoulder by, among others, attachés, assistants and aides-de-camp.
    • Simple aiguillettes are worn by lance corporals of the Household Cavalry and by bandsmen of Dragoon Guards and Dragoon regiments in full dress.

    Aiguillettes are also worn by some police officers.

    • The Commissioner of the Metropolitan Police wears silver aiguillettes in both formal and ceremonial dress.
    • The Commissioner and Assistant Commissioner of the City of London Police wear gold aiguillettes on the right shoulder in ceremonial dress. Some officers of that force also wear gold aiguillettes on the left shoulder in ceremonial dress.

    United States

    Aiguillette
    US Army Service Aiguillette worn on the left arm by aides-de-camp, Army attaches, and assistant Army attaches and on the right arm by aides assigned to the President, White House, First Family, and foreign heads of state with the Class A Uniform and Army Service Uniform
    Aiguillette
    U.S. Navy Ceremonial Guard Dress Aiguillette (White Uniform): one dark blue braided loop and two single loops

    The aiguillette is worn on the right shoulder by military aides to the President and the Vice President. It is worn on the left shoulder by military assistants to the Secretary of Defense and the Secretary of Homeland Security, aides to the Service Secretaries (Secretary of the Army, Secretary of the Navy & Secretary of the Air Force), aide to the NOAA Administrator, military attachés, General Staff Corps officers, and aides to flag officers. The cord colors are gold for the Army and silver for the Air Force, gold and blue for the Coast Guard, Navy and NOAA Commissioned Officer Corps, with one braid “per star” of the Flag Officer (one for RDML, two for RADM, three for VADM and four for ADM), and gold and red for the Marines, with the number of braids corresponding to the rank of the General Officer similar to the Navy use. The gold cord aiguillette is worn by the directors of the United States Marine Band, while the enlisted personnel wear aiguillettes of white cord.

    A red aiguillette is worn on the left shoulder by United States Navy Recruit Division commanders, whereas a blue aiguillette is worn by Recruit Division commanders in training. Recruit Division commanders wearing these are referred colloquially as “red ropes” and “blue ropes” respectively. The U.S. Navy Ceremonial Guard in Washington, D.C. wears a blue aiguillette on summer white uniforms and a white one on winter blue uniforms. From 1983 until the issue of the campaign hat, Women Marine Drill Instructors wore a red aiguillette similar to the Navy.

    A blue aiguillette is worn on the left shoulder by military training leaders at technical training bases. Student airmen leaders at technical training bases (sometimes called “ropes” in reference to the aiguillette) also wear aiguillettes, with green representing the lowest level of student leadership, yellow representing the intermediate level of student leadership, and red representing the highest level of student leadership. Students wearing a white rope are commonly referred to as chapel guides, and are charged with providing social or moral support to their fellow airmen. Airmen who wear the black rope are experts in drill, choir, or ceremonies, and pride themselves on appearance and uniform wear. Airmen who wear a blue and white rope are members of the drum & bugle corps for their base. In the United States Air Force, honor guard members wear a silver aiguillette on the left shoulder.

    Other military cords

    The aiguillette should not be confused with the fourragère, which is used by a number of European militaries as well as by some United States units awarded decorations in the First and Second World Wars.

    Nor should it be confused with the Schützenschnur, a multi-weapon marksmanship decoration awarded by Germany to qualifying soldiers of NATO countries serving in Germany.

    A similar, albeit thicker, device, the Infantry Shoulder Cord, is worn on the right shoulder by all enlisted soldiers and officers in the United States Army whose primary Military Occupational Specialty is infantry.

    Many military units wear dress lanyards.

    References

    1. 1 2
      Picken, Mary Brooks (1957). The fashion dictionary. Funk and Wagnalls. (1973 edition ISBN 0-308-10052-2)
    2. 1 2 3
      Gaimster, David; Hayward, Maria; Mitchell, David; Parker, Karen (2002). “Tudor silver-gilt dress-hooks: a new class of treasure find in England”. Antiquaries Journal. 82: 157–196. doi:10.1017/S0003581500073777. ISSN 0003-5815. S2CID 161564261.
    3. 1 2 3 4
      Scarisbrick, Diana (1995). Tudor and Jacobean jewellery. London: Tate Publishing. ISBN 1-85437-158-4.
    4. ↑ https://en.delachieve.com/axelbant-is-an-accessory-with-a-long-history/
    5. ↑
      “Origin of aiguillettes”. All Hands. 615. Australia: Naval Department Library: 34–35. April 1968.
    6. ↑ The ensuing knots would hang down the shoulder. (As with combat boots, the longer the lace, the less the need to undo the entire lace.) As armour became more ornamental and less practical, so too did the ties. This would also explain the aiguillettes of varying levels of complexity in the uniforms of the Household Cavalry (see picture above), as opposed to other “unarmored” troops.
    7. ↑ A version that says that aiguillettes originated in aides-de-camp and adjutants wearing a pencil at the end of a cord hanging from the shoulder has no historical base.
    8. ↑ « Cette origine est espagnole. Le duc d’Albe, pour se venger de l’abandon d’un corps considérable de Belges, ordonna que les délits qui se commettraient fussent punis de la corde, sans distinction de rang. Ces braves firent dire au duc que, poux faciliter l’exécution de cette mesure, ils porteraient sur le col une corde et un clou. Cette troupe s’étant distinguée, la corde et le clou devinrent dos marques d’honneur, et furent transformés en aiguillette ».

      • Dictionnaire des proverbes français, 3rd edition, by M. de la Mésangère, La Société royale des antiquaires de France, Imprimerie de Crapelet, Treuttel & Würtz, Paris, Strasbuourg & Longdon, 1823

      https://ia600908.us.archive.org/11/items/dictionnairedesp00lam/dictionnairedesp00lam.pdf Public Domain This article incorporates text from this source, which is in the public domain.

    9. ↑ Spain Ministry of Defence. “Aiguillettes”.
    10. 1 2
      Arnold, Janet (1980). Lost from Her Majesties back. The Costume Society. (Items of clothing and jewels lost or given away by Elizabeth I between 1561 and 1583, entered in one of the day books kept for the records of the Wardrobe of Robes.)
    11. ↑ Thomas Thomson, Collection of Inventories (Edinburgh, 1815), p. 278.
    12. ↑
      “Aiguillettes”. Australian Army. 5 October 2016. Archived from the original on 2018-03-19. Retrieved 2019-01-06.
    13. ↑ A-DH-265-000/AG-001 Canadian Armed Forces Dress Instructions, Ch.3, Section 7
    14. ↑ A-DH-265-000/AG-001 Canadian Armed Forces Dress Instructions, Ch. 3 Annex G

    General references

    • Arnold, Janet: Patterns of Fashion: the cut and construction of clothes for men and women 1560–1620, Macmillan 1985. Revised edition 1986. ISBN 0-89676-083-9
    • Arnold, Janet: Queen Elizabeth’s Wardrobe Unlock’d, W S Maney and Son Ltd, Leeds 1988. ISBN 0-901286-20-6
    • Hearn, Karen, ed. Dynasties: Painting in Tudor and Jacobean England 1530–1630. New York: Rizzoli, 1995. ISBN 0-8478-1940-X.


    This article is adapted from “Aiguillette” on Wikipedia, written by its contributors, and used under CC BY-SA 4.0. Text on this page is available under the same licence. Source snapshot: Wikipedia via Kiwix, 2026-07-20.

  • Milnor conjecture (knot theory)

    In knot theory, the Milnor conjecture says that the slice genus of the ( p , q ) {\displaystyle (p,q)} {\displaystyle (p,q)} torus knot is

    ( p − 1 ) ( q − 1 ) 2 . {\displaystyle {\frac {(p-1)(q-1)}{2}}.} {\displaystyle {\frac {(p-1)(q-1)}{2}}.}

    It is in a similar vein to the Thom conjecture.

    The conjecture (named after John Milnor) was first proved by gauge theoretic methods by Peter Kronheimer and Tomasz Mrowka.[1] Jacob Rasmussen later gave a purely combinatorial proof using Khovanov homology, by means of the s-invariant.[2]

    References

    1. ↑ Kronheimer, Peter B.; Mrowka, Tomasz S. (1993), “Gauge theory for embedded surfaces, I” (PDF), Topology, 32 (4): 773–826, doi:10.1016/0040-9383(93)90051-V.
    2. ↑ Rasmussen, Jacob A. (2010). “Khovanov homology and the slice genus”. Inventiones Mathematicae. 182 (2): 419–447. arXiv:math.GT/0402131. doi:10.1007/s00222-010-0275-6.

    This article is adapted from “Milnor conjecture (knot theory)” on Wikipedia, written by its contributors, and used under CC BY-SA 4.0. Text on this page is available under the same licence. Source snapshot: Wikipedia via Kiwix, 2026-07-20.

  • Affine braid group

    In mathematics, an affine braid group is a braid group associated to an affine Coxeter system. Their group rings have quotients called affine Hecke algebras. They are subgroups of double affine braid groups.

    Definition

    References

    • Haiman, Mark (2006), “Cherednik algebras, Macdonald polynomials and combinatorics”, International Congress of Mathematicians, Vol. III, European Mathematical Society, Zürich, pp. 843–872, ISBN 978-3-03719-022-7, MR 2275709, archived from the original on 2011-08-20, retrieved 2011-06-10
    • Macdonald, I. G. Affine Hecke Algebras and Orthogonal Polynomials. Cambridge Tracts in Mathematics, 157. Cambridge University Press, Cambridge, Eng., 2003. x+175 pp. ISBN 0-521-82472-9 MR 1976581

    This article is adapted from “Affine braid group” on Wikipedia, written by its contributors, and used under CC BY-SA 4.0. Text on this page is available under the same licence. Source snapshot: Wikipedia via Kiwix, 2026-07-20.

  • Adjustable grip hitch

    Adjustable grip hitch
    Adjustable grip hitch
    Names Adjustable grip hitch, Adjustable loop, Adjustable hitch, Adjustable bend, Adjustable Cawley Hitch, Cawley Hitch
    Category Hitch
    Related Taut-line hitch, Farrimond friction hitch
    Releasing Non-jamming
    Typical use Keep a rope taut
    ABoK –

    The adjustable grip hitch is a simple and useful friction hitch which may easily be shifted up and down the rope while slack. It will hold fast when loaded, but slip when shock loaded until tension is relieved enough for it to again hold fast. It serves the same purpose as the taut-line hitch, e.g. tensioning a tent’s guy line.

    This knot is also called the adjustable loop[1] and Cawley adjustable hitch. It was conceived in 1982 by Canadian climber Robert Chisnall.[2]

    Tying

    Adjustable grip hitch
    Tying the adjustable grip hitch (slipped)

    The working end is wrapped inwards around the standing part (A-B) twice (1). Then another turn is made around both parts and a bight is pulled through the last wrap (2, 3) for the slipped version (left image), or just the end for the non-slipped version (right image). The knot needs to be pulled tight to actually grip (the slack is pulled out of the windings and the knot pulled tight at C and D).
    By pushing the knot along the standing part in direction of A, the line can be tightened.

    The grip can be improved by adding a third turn around the standing part.

    The slipped adjustable grip hitch can be easily untied by pulling the end E, even under quite heavy load.

    Adjustable Bend

    Adjustable grip hitch
    adjustable bend

    Using the same knot core an adjustable bend can be made to join two lines. For that the knot is tied in both working ends, each around the other line’s standing part.[2] The length of the line can be then adjusted by pushing the knots together or apart.

    Alternatives

    External links

    See also

    References

    1. ↑ Budworth, Geoffrey (2003). The Ultimate Encyclopedia of Knots & Ropework. Hermes House. ISBN 978-1-84309-138-7.
    2. 1 2 Budworth, Geoffrey (1997). The Complete Book of Knots. The Lyons Press. p. 95. ISBN 1-55821-632-4.

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  • Miller’s knot

    Miller’s knot
    Miller's knot

    ABoK #1241 around the neck of a paper bag
    Names Miller’s knot, sack knot, bag knot
    Category Binding
    Related Ground-line hitch, constrictor knot, strangle knot, clove hitch, Bottle sling
    ABoK #388, #389, #390, #1241, #1242, #1243, #1244, #1674, #2007, #2186, #2300, #2554

    A miller’s knot (also sack knot or bag knot) is a binding knot used to secure the opening of a sack or bag. Historically, large sacks often contained grains; thus the association of these knots with the miller’s trade. Several knots are known interchangeably by these three names.[1]

    Variations

    Several other distinct knots have historically been known as miller’s, sack, or bag knots; namely ABOK-1241, ABOK-1242, ABOK-1243 (ground-line hitch), ABOK-1674, ABOK-11 and their slipped versions. These fit the short description “two crossing turns – ends tucked under”.

    The following 3 knots do not fit the simple “two crossing turns – ends tucked under” definition but are very secure:

    • A slipped constrictor knot where the ends are passed through the opposing slips for security is quite secure as a bag knot. Unslipped, it is even more secure, but it may have to be cut at the bridge to open the bag.
    • A slipped strangle knot where the ends are passed through the opposing slips for security is also quite secure as a bag knot.
    • A bottle sling around a swirled and folded neck of the bag (elbow of the neck) is also a very solid and reliable bag knot. The fold provides the slight bulge a bottle sling needs for a secure hold. The bag may be hanged fully loaded by the ends of the bottle sling tied together.

    Constrictor knot makes a fiercer binding knot, but miller’s knot is suitable for most applications, and is easier to tie and untie. Miller’s knot makes a great hitch, like the similar ground line hitch. Binding usage has force emanating from inside rope ring evenly, hitch usage has force input from one side, then reducing around.

    Tying

    The common aspects of the most common bag knots are two crossing turns, and both ends tucked under some turns near the crossing point. Two ends, and two turns one can tuck under, gives a limited number of alternatives. All of these knots can also be made in a slipped form by starting with a bight and/or by completing the final tuck with a bight instead of the end.[2]

    To avoid ambiguity, versions of these knots that are not slipped are pictured below with the reference numbers found in The Ashley Book of Knots.

    This is to tie a constrictor knot version of the miller’s knot:

    1 Constrictor knot
    1 Constrictor knot
    1. Grip the neck of the bag with the left hand,
    2. Fix / immobilize one end of the rope tucked upwards over the left hand long finger and under the index finger (option 1 : double folded as a bight to prepare a start-side-slip for the final knot)
    3. Make two crossing turns around the neck of the bag. Detailed steps:
      1. Cross over the hand downwards and take one turn around (front, then back) the neck of the bag at the sack side of the hand (under the hand)
      2. Cross over the hand upwards as well as over the immobilized other end,
      3. Take a second turn around the neck of the bag at the opening side of the hand (over the hand)
    4. Cross over the immobilized other end of the rope, then cross back tucking under the crossing point of step 3.2 (of the immobilized other end, and the part between the two turns), (option 2 : this last tuck with a bight instead of the end as an end-side-slip for the final knot).

    To tie the other variants:

    1. If at the last step one chooses not to cross the immobilized other end and tuck only inwards under the part between the two turns, the knot will be an ABoK #11 or clove hitch.
    2. If at the last step one chooses to cross the immobilized other end and tuck only under the part between the two turns, the knot will be an ABoK #1242 (tuck inwards)
    3. or a ground-line hitch (ABoK #1243 – tuck outwards) pictured. It should be tightened by pulling the end first. It is also called spar hitch
    4. If at the last step one chooses to cross over the crossing point, and then tuck outwards under the first turn, the knot will be an ABoK #1241 pictured.
    5. If at the last step one chooses to cross over the crossing point, and then tuck inwards under the first turn, the knot will be an ABoK #1674 pictured. Shown in a slipped form at entry #1244, this variation is noted by Ashley as having better binding characteristics than the others.[2]

    Tying other knots that also may function very well as a bag knot but are slightly different from above descriptions:

    1. Strangle knot (two non crossing turns with one end crossing over both and tucked under both)
      1. Grip the neck of the bag with the left hand,
      2. Fix / immobilize one end of the rope tucked upwards over the left hand long and under the index finger (option 1 : double folded as a bight to prepare a start-side-slip for the final knot)
      3. Make two parallel turns around the neck of the bag. Detailed steps:
        1. Cross over the hand downwards and take one turn around (front, then back) the neck of the bag first at the sack side of the hand (under the hand), then over the hand and behind the immobilized other end.
        2. Take another turn crossing over the immobilized other end, crossing downwards over the hand around (front, then back) the neck of the bag parallel to the first turn
      4. Cross over the immobilized other end of the rope, then cross back tucking under both turns, (option 2 : this last tuck with a bight instead of the end as an end-side-slip for the final knot).
    2. Bottle sling (sides of a bight form one turn each, pleat between all 4 enter and exit points inside of two turns)
    • 8-1- For ABOK #1142 bottle sling start with an overhand knot pulled to a slip knot
      8-1- For ABOK #1142 bottle sling start with an overhand knot pulled to a slip knot
    • 8-2-The end side of the slip is folded into a loop with end side up
      8-2-The end side of the slip is folded into a loop with end side up
    • 8-3-The resulting loop is folded under and the slip side of the overhand knot pulled through it
      8-3-The resulting loop is folded under and the slip side of the overhand knot pulled through it
    • 8-4-The remaining slip is passed over the loop and under the knot bight pulled through the loop
      8-4-The remaining slip is passed over the loop and under the knot bight pulled through the loop
    • 8-5-The hole in the middle ready to be slipped over the swirled and double folded bag mouth, and tightened
      8-5-The hole in the middle ready to be slipped over the swirled and double folded bag mouth, and tightened

    See also

    References

    1. ↑ Clifford W. Ashley, The Ashley Book of Knots (New York: Doubleday, 1944), 62.
    2. 1 2 Ashley, 224.

    This article is adapted from “Miller's knot” on Wikipedia, written by its contributors, and used under CC BY-SA 4.0. Text on this page is available under the same licence. Source snapshot: Wikipedia via Kiwix, 2026-07-20.

  • Adjustable bend

    Adjustable bend
    Adjustable bend

    An adjustable bend, joining two kernmantle ropes
    Category Bend
    Related rolling hitch, taut-line hitch
    Releasing Non-jamming
    Typical use Climbing
    ABoK 1472, 2558, 2031

    The adjustable bend[1] is a bend knot that is easy to lengthen or shorten.

    A rolling hitch is used to tie the end of each rope to the standing part of the other. Clifford Ashley suggested it for tying guy ropes.

    References

    1. ↑ The complete guide to knots and knot tying — Geoffrey Budworth — p.52 — ISBN 0-7548-0422-4

    See also


    This article is adapted from “Adjustable bend” on Wikipedia, written by its contributors, and used under CC BY-SA 4.0. Text on this page is available under the same licence. Source snapshot: Wikipedia via Kiwix, 2026-07-20.

  • Merovingian knot

    Merovingian knot
    The Merovingian knot on a solid red necktie.

    The Merovingian knot, or the Ediety knot, is a necktie knot characterized by its inverted structure and layered appearance. It gained popular recognition after being worn by the character known as the Merovingian in the film The Matrix Reloaded, part of The Matrix franchise. The knot is notable for its unconventional tying method, in which the narrow end of the tie is used to construct the knot rather than the wide end.

    History

    The knot became more known following the 2003 release of The Matrix Reloaded, where it was worn by the character the Merovingian. Although sometimes described as a modern variation of the Windsor knot, the Merovingian knot is structurally distinct due to its reversed tying technique and its use of the narrow blade as the active end.[1]

    Characteristics

    The Merovingian knot is distinguished by:

    1. An inside-out construction method
    2. A narrow end that remains visible and wraps around the front
    3. A layered, slightly asymmetrical appearance
    4. A relatively small, compact knot size

    Tying method

    To tie the Merovingian knot:

    Drape the tie around the neck with the wide end shorter than the narrow end.[1]

    1. Cross the narrow end over the wide end.
    2. Wrap the narrow end behind and through the neck loop.
    3. Pass the narrow end across the front and up through the loop again.
    4. Tuck the narrow end downward through the front horizontal band.
    5. Adjust and tighten carefully to shape the knot.

    Because the knot is tied with the narrow end, additional adjustment may be required to ensure the wide blade reaches the proper length.[2]

    References

    External links

    This article is adapted from “Merovingian knot” on Wikipedia, written by its contributors, and used under CC BY-SA 4.0. Text on this page is available under the same licence. Source snapshot: Wikipedia via Kiwix, 2026-07-20.

  • Matthew Walker knot

    Matthew Walker knot
    Matthew Walker knot
    Category Decorative
    ABoK 678
    Instructions

    A Matthew Walker knot is a decorative knot that is used to keep the end of a rope from fraying. It is tied by unraveling the strands of a twisted rope, knotting the strands together, then laying up the strands together again. It may also be tied using several separate cords, in which case it keeps the cords together in a bundle. The traditional use of the knot is to form a knob or “stopper” to prevent the end of the rope from passing through a hole, for instance in rigging the lanyards which tension the shrouds on older sailing ships with standing rigging of fibre cordage.

    It is not specifically known who Matthew Walker was, nor why this knot was named after him. However, early references from the 19th century suggest he may have been a ship’s rigger in the Royal Navy.

    The following quote from The Ashley Book of Knots gives possible origins of the knot:[1]

    The FULL or DOUBLE MATTHEW WALKER KNOT.

    Lever in 1808 speaks of “MATTHEW WALKER’S KNOT” and describes the knot which Alston in 1860 calls the “DOUBLE MATTHEW WALKER KNOT.” A refinement of the original knot had in the meantime taken over the original name, which is now generally modified to “a MATTHEW WALKER”.

    Lever’s familiar expression, “MATTHEW WALKER’S KNOT,” suggests that he may have known the inventor, who was possibly a master rigger in one of the British naval dockyards. Many myths have grown up around Matthew Walker, “the only man ever to have a knot named for him.” Dr. Frederic Lucas, of the American Museum of Natural History, once told me the following story of the Origin of the knot, which he had heard off the Chincha Islands while loading guano in 1869.

    A sailor, having been sentenced to death by a judge who in earlier life had been a sailor himself, was reprieved by the judge because of their common fellowship of the sea. The judge offered the sailor a full pardon if he could show him a knot that he, the judge, could neither tie nor untie.

    The sailor called for ten fathoms of rope and, having retired to the privacy of his cell, unlaid the rope halfway, put in a MATTHEW WALKER KNOT, and then laid up the rope again to the end.

    So Matthew Walker secured his pardon, and the world gained an excellent knot.

    This knot is highly decorative, and was historically one of the most common and important knots. On a modern yacht, it is almost unused and unknown.[2]

    It has been used in making stopper knots where lariats are used.[3]

    How to tie

    Matthew Walker knot

    A Matthew Walker knot is tied in a circular bundle of any number of strands. To tie the knot, the tier takes each strand and forms a loop around the rest of the bundle, then passes the end through the newly formed loop to form an overhand knot. They then move to the next strand over, moving around the bundle in the direction they pass the loops. Tying the first strand around the bundle is straightforward, but each subsequent end must be passed through the previously-formed loops in order to contain all of the other strands in its loop. When tightening, it may help to roll the knot along the bundle, especially when using only two strands. The final effect is a spiral knot vaguely resembling a section of a barber’s pole.

    • Matthew Walker knot
    • Matthew Walker knot

    See also

    References

    1. ↑ Ashley, Clifford (1944). The Ashley Book of Knots, p.118. ISBN 9780385040259.
    2. ↑ “Double Matthew Walker Knot” (animation). animated knots by Grog. Retrieved March 24, 2013.
    3. ↑ “How To… Tie A Stopper Knot”. Cowboy Way. Retrieved March 24, 2013.

    Knob knots

    External links

    This article is adapted from “Matthew Walker knot” on Wikipedia, written by its contributors, and used under CC BY-SA 4.0. Text on this page is available under the same licence. Source snapshot: Wikipedia via Kiwix, 2026-07-20.

  • 7 2 knot

    Pentatwist knot, five-twist knot
    7 2 knot
    Arf invariant 1
    Braid length 9
    Braid no. 4
    Bridge no. 2
    Crosscap no. 2
    Crossing no. 7
    Genus 1
    Hyperbolic volume 2.82812
    Stick no. 9
    Unknotting no. 1
    Conway notation [52]
    A–B notation 72
    Dowker notation 4, 8, 12, 14, 2, 6, 10
    Last / Next 71 / 73
    Other
    alternating, prime, reversible

    In knot theory, the Pentatwist knot, also known as the five-twist knot, or the 72, is one of seven prime knots with crossing number seven. It is the fifth twist knot.

    Invariants

    Its Alexander polynomial is

    Δ ( t ) = 3 t − 5 + 3 t − 1 , {\displaystyle \Delta (t)=3t-5+3t^{-1},\,} {\displaystyle \Delta (t)=3t-5+3t^{-1},\,}

    its Conway polynomial is

    ∇ ( z ) = 3 z 2 + 1 , {\displaystyle \nabla (z)=3z^{2}+1,\,} {\displaystyle \nabla (z)=3z^{2}+1,\,}

    and its Jones polynomial is

    V ( q ) = − q − 8 + q − 7 − q − 6 + 2 q − 5 − 2 q − 4 + 2 q − 3 − q − 2 + q − 1 . {\displaystyle V(q)=-q^{-8}+q^{-7}-q^{-6}+2q^{-5}-2q^{-4}+2q^{-3}-q^{-2}+q^{-1}.\,} {\displaystyle V(q)=-q^{-8}+q^{-7}-q^{-6}+2q^{-5}-2q^{-4}+2q^{-3}-q^{-2}+q^{-1}.\,}[1]

    Example

    References

    This article is adapted from “7 2 knot” on Wikipedia, written by its contributors, and used under CC BY-SA 4.0. Text on this page is available under the same licence. Source snapshot: Wikipedia via Kiwix, 2026-07-20.