Category: Knots

In knot theory, the self-linking number is an invariant of framed knots. It is related to the linking number of curves.

A framing of a knot is a choice of a non-zero non-tangent vector at each point of the knot. More precisely, a framing is a choice of a non-zero section in the normal bundle of the knot, i.e. a (non-zero) normal vector field. Given a framed knot C, the self-linking number is defined to be the linking number of C with a new curve obtained by pushing points of C along the framing vectors.

Given a Seifert surface for a knot, the associated Seifert framing is obtained by taking a tangent vector to the surface pointing inwards and perpendicular to the knot. The self-linking number obtained from a Seifert framing is always zero.^[1]

The blackboard framing of a knot is the framing where each of the vectors points in the vertical (z) direction. The self-linking number obtained from the blackboard framing is called the Kauffman self-linking number of the knot. This is not a knot invariant because it is only well-defined up to regular isotopy.

• Chernov, Vladimir (2005), “Framed knots in 3-manifolds and affine self-linking numbers”, Journal of Knot Theory and its Ramifications, 14 (6): 791–818, arXiv:math/0105139, doi:10.1142/S0218216505004056, MR 2172898.
• Moskovich, Daniel (2004), “Framing and the self-linking integral”, Far East Journal of Mathematical Sciences, 14 (2): 165–183, arXiv:math/0211223, Bibcode:2002math…..11223M, MR 2105976

Carrick mat
Carrick mat made with three passes
Names	Carrick mat, Prolong knot
Category	Decorative
Related	Carrick bend, Turk’s head, Austrian knot
Typical use	Mat for padding or decoration
ABoK	#2242, #2244
Instructions

818 knot
Basic seamless form
Arf invariant	1
Braid length	8
Braid no.	3
Bridge no.	3
Crosscap no.	4
Crossing no.	8
Genus	3
Hyperbolic volume	12.35090621
Unknotting no.	2
Conway notation	[8*]
A–B notation	818
Dowker notation	6, 8, 10, 12, 14, 16, 2, 4
D–T notation	8a12
Last / Next	817 / 819
Other
alternating, prime, fibered, prime, fully amphichiral

The carrick mat is a flat woven decorative knot which can be used as a mat or pad.^[1] Its name is based on the mat’s decorative-type carrick bend with the ends connected together, forming an endless knot. A larger form, called the prolong knot, is made by expanding the basic carrick mat by extending, twisting, and overlapping its outer bights, then weaving the free ends through them. This process may be repeated to produce an arbitrarily long mat.^[2]

In its basic form it is the same as a 3-lead, 4-bight Turk’s head knot.^[3] The basic carrick mat, made with two passes of rope, also forms the central motif in the logo of the International Guild of Knot Tyers.^[4]

When tied to form a cylinder around the central opening, instead of lying flat, it can be used as a woggle.

• List of knots

• “8_18“, The Knot Atlas.

Carrick bend loop
Names	Carrick bend loop, bollard loop[1]
ABoK	#1033

A carrick loop^[2] is a knot used to make a reliable and stable loop at the end of a rope formed by the tail turned around and attached to the main part using a carrick bend.

The carrick bend knot closing the carrick bend loop will consist of

• a simple closed loop of the main part and
• a small closed loop of the rope tail
• woven together in a basket weave pattern and then
• tightened.

The knot will have 8 crossing points, 7 holes, the central hole with 4 straight edges, and the 6 others with 2 straight and one external curved edge. The main part, the end, and the two loop edges will enter the knot from the four corners of the knot when tied flat; If the ones that will carry the heaviest loads are placed at opposite corners the knot will hold better.

The carrick loop is reliable, and easy to untie, but there are no advantages over other loops that are easier to tie.^[2]

There are two possible variations depending on the loop ends angle of entry to the knot relative to each other, roughly 90 degrees, or 180 degrees; The former giving a more flat and decorative knot and the latter, double coin knot, being more secure.

• List of knots

Seizings are a class of stopping knots used to semi-permanently bind together two ropes, two parts of the same rope, or rope and another object.^[1] Akin to lashings, they use string or small-stuff to produce friction and leverage to immobilize larger ropes. Seizings are not recommended for heavy loads for critical use as strain reduces the diameter of the main rope and can permit slippage even with proper construction. According to The Ashley Book of Knots, “A seizing holds several objects together.”^[1] The other type of stopping knots are whipping knots.

A throat seizing is a seized round turn. It is used when turning in deadeyes, and has riding turns but no crossing turns. The end of the stay or shroud should first be stopped around the deadeye. — The Ashley Book of Knots^[1]

• Ropework

Carrick bend
Top – A fully interwoven diagonally opposed carrick bend Bottom – capsized
Names	Carrick bend, Double carrick bend, Double coin knot, Ten accord knot, Bosun’s knot, Basketweave knot, Chinese knot, Josephine knot, Whistle lanyard, Sailor’s breastplate knot, Sailors knot, Pretzel knot, Wake knot
Category	Bend
Related	Single carrick bend, Diamond knot, Carrick mat
Releasing	Non-jamming
ABoK	#1428, #1439

The Carrick bend, also known as the Sailor’s breastplate, is a knot used for joining two lines. It is particularly appropriate for very heavy rope or cable that is too large and stiff to be easily formed into other common bends.^[1][2] It will not jam even after carrying a significant load or being soaked with water.^[3]

As with many other members of the basket weave knot family, the carrick bend’s aesthetically pleasing interwoven and symmetrical shape has also made it popular for decorative purposes.

The Carrick bend is known as the “Wake knot” or “Ormonde knot” when it is used as a heraldic badge.^[4]

This knot’s name dates back to at least 1783, when it was included in a nautical bilingual dictionary authored by Daniel Lescallier.^[1][5] Its origins prior to that are not known with certainty. There are several possible explanations for the name “Carrick” being associated with this bend. The Elizabethan era plasterwork of Ormonde Castle in Carrick-on-Suir shows numerous carrick bends molded^[6] in relief. Or the name may come from Carrick Roads—a large natural anchorage by Falmouth in Cornwall, England. The name may also have been derived from the Carrack, a medieval type of ship.^[7]

The eight crossings within the carrick bend allow for many similar-looking knots to be made. The lines in a “full” or “true” carrick bend alternate between over and under at every crossing. There are also two ways the ends can emerge from the knot: diagonally opposed or from the same side. The latter form is also called the double coin knot. The form with the ends emerging diagonally opposed is considered more secure.^[1]

Unfortunately, with so many permutations, the carrick bend is prone to being tied incorrectly.^[3]

The carrick bend, also called full carrick bend, sailor’s knot, and anchor bend, is perhaps the nearest thing we have to a perfect bend. It is symmetrical, it is easy to tie, it does not slip easily in wet material, it is among the strongest of knots, it cannot jam and is readily untied. To offset this array of excellencies is the sole objection that it is somewhat bulky.

Capsized

The carrick bend is generally tied in a flat interwoven form as shown above. Without additional measures it will collapse into a different shape when tightened, a process known as capsizing, with the degree of capsizing depending on the looseness of the weave. This capsized form is both secure and stable once tightened, although it is bulkier than the seized form below. Incomplete capsizing resulting from a tight weave produces a form that is likewise secure and stable, but which is more difficult to untie, countering one of the advantages of the carrick bend. When the knot is allowed to capsize naturally under tension, considerable slippage of line through the knot can occur before tightening, so the knot should be set carefully before loading to avoid this slippage in use.^[9]

Seized

In the interest of making the carrick bend easier to untie, especially when tied in extremely large rope, the ends may be seized to prevent the knot from collapsing when load is applied. This practice also keeps the knot’s profile flatter and can ease its passage over capstans or winches.^[10]

The ends are traditionally seized to their standing part using a round seizing. For expediency, a series of double constrictor knots, drawn very tight, may also be used.^[2] When seizing the carrick bend, both ends must be secured to their standing parts or the bend will slip.

In the decorative variation, both standing ends enter from one side and both working ends exit from the other. In this configuration, the knot is known as the Josephine knot (macrame) or double coin knot (Chinese knotting). This form of the carrick bend is found depicted in heraldry, sometimes with the tails of heraldic serpents woven (or “nowed”) into this knot.^[11] In heraldry, the knot is associated with Hereward the Wake and is known under the name Wake knot.^[7] It is depicted in the coat of arms of Bourne Town Council, Lincolnshire.^[12]

The knot can be tied using doubled lines for an even flatter, more elaborate appearance. A doubled carrick bend was used to ornamentally secure the lanyards on the breastplate of the US Navy Mark V diving helmet during inspection and between dives.^[13][14]

When the ends of the carrick bend are connected together, or more practically hidden behind the knot, it becomes a carrick mat. This same configuration is also one of the most basic Turk’s head knots.

The fully interwoven diagonal carrick bend is the most secure variation. All other forms are inferior^[3] and not recommended as bends.^[1]

Although the carrick bend has a reputation for strength, some tests have shown it to be as weak as 65% efficiency.^[1]

• List of bend knots
• List of knots

• Grog. “How to tie the carrick bend”. Animated Knots.
• “L8a8 (mathematical “link” topologically equivalent to seized carrick bend of practical knot-tying)“, The Knot Atlas.
• Animated carrick bend video and step by step process.

In mathematics, a Seifert surface (named after German mathematician Herbert Seifert^[1][2]) is an orientable surface whose boundary is a given knot or link.

Such surfaces can be used to study the properties of the associated knot or link. For example, many knot invariants are most easily calculated using a Seifert surface. Seifert surfaces are also interesting in their own right, and the subject of considerable research.

Specifically, let L be a tame oriented knot or link in Euclidean 3-space (or in the 3-sphere). A Seifert surface is a compact, connected, oriented surface S embedded in 3-space whose boundary is L such that the orientation on L is just the induced orientation from S.

Note that any compact, connected, oriented surface with nonempty boundary in Euclidean 3-space is the Seifert surface associated to its boundary link. A single knot or link can have many different inequivalent Seifert surfaces. A Seifert surface must be oriented. It is possible to associate surfaces to knots which are neither oriented nor orientable as well.

The standard Möbius strip has the unknot for a boundary but is not a Seifert surface for the unknot because it is not orientable.

The “checkerboard” coloring of the usual minimal crossing projection of the trefoil knot gives a Mobius strip with three half twists. As with the previous example, this is not a Seifert surface as it is not orientable. Applying Seifert’s algorithm to this diagram, as expected, does produce a Seifert surface; in this case, it is a punctured torus of genus g = 1, and the Seifert matrix is

${\displaystyle V={\begin{pmatrix}1&-1\\0&1\end{pmatrix}}.}$

It is a theorem that any link always has an associated Seifert surface. This theorem was first published by Frankl and Pontryagin in 1930.^[3] A different proof was published in 1934 by Herbert Seifert and relies on what is now called the Seifert algorithm. The algorithm produces a Seifert surface ${\displaystyle S}$, given a projection of the knot or link in question.

Suppose that link has m components (m = 1 for a knot), the diagram has d crossing points, and resolving the crossings (preserving the orientation of the knot) yields f circles. Then the surface ${\displaystyle S}$ is constructed from f disjoint disks by attaching d bands. The homology group ${\displaystyle H_{1}(S)}$ is free abelian on 2g generators, where

${\displaystyle g={\frac {1}{2}}(2+d-f-m)}$

is the genus of ${\displaystyle S}$. The intersection form Q on ${\displaystyle H_{1}(S)}$ is skew-symmetric, and there is a basis of 2g cycles ${\displaystyle a_{1},a_{2},\ldots ,a_{2g}}$ with
${\displaystyle Q=(Q(a_{i},a_{j}))}$ equal to a direct sum of the g copies of the matrix

${\displaystyle {\begin{pmatrix}0&-1\\1&0\end{pmatrix}}}$

The 2g × 2g integer Seifert matrix

${\displaystyle V=(v(i,j))}$

has ${\displaystyle v(i,j)}$ the linking number in Euclidean 3-space (or in the 3-sphere) of a_i and the “pushoff” of a_j in the positive direction of ${\displaystyle S}$. More precisely, recalling that Seifert surfaces are bicollared, meaning that we can extend the embedding of ${\displaystyle S}$ to an embedding of ${\displaystyle S\times [-1,1]}$, given some representative loop ${\displaystyle x}$ which is homology generator in the interior of ${\displaystyle S}$, the positive pushout is ${\displaystyle x\times \{1\}}$ and the negative pushout is ${\displaystyle x\times \{-1\}}$.^[4]

With this, we have

${\displaystyle V-V^{*}=Q,}$

where V^∗ = (v(j, i)) the transpose matrix. Every integer 2g × 2g matrix ${\displaystyle V}$ with ${\displaystyle V-V^{*}=Q}$ arises as the Seifert matrix of a knot with genus g Seifert surface.

The Alexander polynomial is computed from the Seifert matrix by ${\displaystyle A(t)=\det \left(V-tV^{*}\right),}$ which is a polynomial of degree at most 2g in the indeterminate ${\displaystyle t.}$ The Alexander polynomial is independent of the choice of Seifert surface ${\displaystyle S,}$ and is an invariant of the knot or link.

The signature of a knot is the signature of the symmetric Seifert matrix ${\displaystyle V+V^{\mathrm {T} }.}$ It is again an invariant of the knot or link.

Seifert surfaces are not at all unique: a Seifert surface S of genus g and Seifert matrix V can be modified by a topological surgery, resulting in a Seifert surface S′ of genus g + 1 and Seifert matrix

${\displaystyle V’=V\oplus {\begin{pmatrix}0&1\\1&0\end{pmatrix}}.}$

The genus of a knot K is the knot invariant defined by the minimal genus g of a Seifert surface for K.

For instance:

• An unknot—which is, by definition, the boundary of a disc—has genus zero. Moreover, the unknot is the only knot with genus zero.
• The trefoil knot has genus 1, as does the figure-eight knot.
• The genus of a (p, q)-torus knot is (p − 1)(q − 1)/2
• The degree of a knot’s Alexander polynomial is a lower bound on twice its genus.

A fundamental property of the genus is that it is additive with respect to the knot sum:

${\displaystyle g(K_{1}\mathbin {\#} K_{2})=g(K_{1})+g(K_{2})}$

In general, the genus of a knot is difficult to compute, and the Seifert algorithm usually does not produce a Seifert surface of least genus. For this reason other related invariants are sometimes useful. The canonical genus ${\displaystyle g_{c}}$ of a knot is the least genus of all Seifert surfaces that can be constructed by the Seifert algorithm, and the free genus ${\displaystyle g_{f}}$ is the least genus of all Seifert surfaces whose complement in ${\displaystyle S^{3}}$ is a handlebody. (The complement of a Seifert surface generated by the Seifert algorithm is always a handlebody.) For any knot the inequality ${\displaystyle g\leq g_{f}\leq g_{c}}$ obviously holds, so in particular these invariants place upper bounds on the genus.^[5]

The knot genus is NP-complete by work of Ian Agol, Joel Hass and William Thurston.^[6]

It has been shown that there are Seifert surfaces of the same genus that do not become isotopic either topologically or smoothly in the 4-ball.^[7][8]

• Crosscap number
• Arf invariant of a knot
• Murasugi sum
• Slice genus

• The SeifertView programme of Jack van Wijk visualizes the Seifert surfaces of knots constructed using Seifert’s algorithm.

A carpet page is a full page in an illuminated manuscript containing intricate, non-figurative, patterned designs.^[1] They are a characteristic feature of Insular manuscripts, and typically placed at the beginning of a Gospel Book. Carpet pages are characterised by mainly geometrical ornamentation which may include repeated animal forms. They are distinct from pages devoted to highly decorated historiated initials, though the style of decoration may be very similar.^[2]

Carpet pages are characterised by ornamentation with brilliant colors, active lines and complex patterns of interlace. They are normally symmetrical, or very nearly so, about both a horizontal and vertical axis, though for example the pictured page from the Lindisfarne Gospels is only symmetrical about a vertical axis. Some art historians find their origin in similar Coptic decorative book pages,^[3] and they also clearly borrow from contemporary metalwork decoration. Oriental carpets, or other textiles, may themselves have been influences. The tooled leather book binding of the St Cuthbert Gospel represents a simple carpet page in another medium,^[4] and the few surviving treasure bindings – metalwork book covers or book shrines – from the same period, such as that on the Lindau Gospels, are also close parallels.^[5] Roman floor mosaics seen in post-Roman Britain, are also cited as a possible source.^[6] The Hebrew Codex Cairensis, from 9th century Galilee, also contains a similar type of page, but stylistically very different.

The earliest surviving example is from the early 7th-century Bobbio Orosius, and relates more closely to Late Antique decoration. There are notable carpet pages in the Book of Kells, the Lindisfarne Gospels, the Book of Durrow, and other manuscripts.^[7]

Carpet pages are also found in some medieval Hebrew manuscripts, typically opening the major sections of the book. Islamic manuscripts, especially Qur’ans, often have pages entirely devoted to complex geometrical decoration, but the term is not usually used of them.

• 
  Early insular example from the Book of Durrow
• 
  Carpet page from the Book of Kells
• 
  Page from a Qur’an manuscript, 1182
• 
  Page from a Qur’an manuscript, c. 1370
• 
  Page from a Qur’an by Ibn al-Bawwab, 1001 AD
• 
  Carpet page from the Leningrad Codex

• Treasures of early Irish art, 1500 B.C. to 1500 A.D.: from the collections of the National Museum of Ireland, Royal Irish Academy, Trinity College, Dublin, an exhibition catalogue from The Metropolitan Museum of Art (fully available online as PDF), which contains material on carpet pages
• “Sultan Barquq’s Qur’an”. British Library.
• Howie, Elizabeth. “Dublin, Trinity College MS A.4.5 (57) — Gospel Book (Book of Durrow)”. The University of North Carolina at Chapel Hill. Archived from the original on 11 June 2017.
• “Leningrad Carpet Page”. West Semitic Research Project. Archived from the original on 2010-11-26.

Scoubidou (Craftlace, scoobies, lanyard, gimp, or boondoggle) is material used in knotting craft. It originated in France, where it became a fad in the late 1950s and has lost popularity. It is named after the 1958 song of the same name as sung by the French singer Sacha Distel.

Scoubidou returned to fashion in various countries in the 1980s, and later in 2004 and 2005. It uses commercially supplied plastic strips or tubes.^[1]

The most common kind of thread used for the craft is flat and comes in many colors, sometimes called “lanyard” or “gimp thread”, often depending on region. Another kind of scoubidou thread is supple, round, and hollow plasticized PVC tubes usually about 80 centimetres in length. They are sold in various colors, sizes, and types, and are used to make items by binding them together with knots. On account of their elasticity and hollow cross-section—which enables them to collapse and deform when pulled—they form tight and stable knots. Key chains, friendship bands and other trinkets are most commonly woven, although more complicated shapes and figures can also be created.^[2]

Most of the knots used in scoubidou were already used in bast fibre, while the creations possible with scoubidou are similar to traditional corn dollies and macrame.

Creations such as dragons, Eiffel Towers, bridges, and birds can be made by putting wires inside the stitch to keep it stable, and to enable bending the stitch and keeping it bent.

• Various knot types
• 
  Cobra in the foreground, double box / double barrel in the rear

• Lucet

Button knot
Names	Button knot, Stopper knots, As numbered in picture : 1- Fiador, 2- Sailor’s diamond (#693), 3- Figure-eight loop, 4- Diamond, 5- knife lanyard, 6- Chinese button, 7- Chinese button doubled, 8- True lover’s, 9- Ashley’s, 10- Celtic button, 11- Celtic button on the bight (and thus doubled and with lanyard loop), 12- Friendship, 13- Figure-eight, 14- Overhand
Typical use	Keeps the line from slipping out of things.

A button knot is a knot that forms a bulge of thread. Button knots are essentially stopper knots, but may be aesthetically pleasing enough to be used as a button on clothes.

The single-strand button is a third type of knob knot, in which the working end leaves the knot at the neck, parallel with the standing part, so that the two parts, or ends, together form a stem. The lay of the two ends is the same, and the knot is symmetrical throughout. — The Ashley Book of Knots^[1]

There are many methods for tying button knots, such as the Chinese button knot, the Celtic button knot and the monkey fist. The Ashley Book of Knots contains over a hundred examples.

• 
  Celtic button knot single, with 4 steps of tying it
• 
  Celtic button knot double, flat and tightened forms
• 
  Celtic button knot formed on the bight, untightened – flat
• 
  Celtic button knot formed on the bight, tightened
• 
  Chinese button knot front side
• 
  Chinese button knot back side
• 
  Rose knot on 2 ropes step1 wall
• 
  Rose knot on 2 ropes step2 crown
• 
  Rose knot on 2 ropes step3
• 
  Rose knot on 2 ropes step4 second wall
• 
  Rose knot on 2 ropes step5
• 
  Rose knot on 2 ropes step6 second crown and tucked ends
• 
  Rose knot on 2 ropes top view
• 
  Rose knot on 2 ropes side view
• 
  Rose knot on 2 ropes back view, animated tying
• 
  Rose knot on 4 ropes
• 
  tying of Rose knot on 4 ropes
• 
  Pairs of monkey’s fists formed as cufflinks in silk
• 
  a larger monkey’s fist formed with a thicker rope
• 
  a heavy monkey’s fist for throwing with an eye splice

Savoy knot
The Savoy badge.
Information
Family	House of Savoy
Region	Savoy
Note(s): The first to bear the knot is said to have been Thomas I, Count of Savoy, in the 13th century.[1]

The Savoy knot, a type of decorative knot, is a heraldic knot used primarily in Italian heraldry. It is most notable for its appearance on the heraldic badge of the House of Savoy, where it is accompanied by the motto Stringe ma non costringe, “It tightens, but does not constrain”.^[2] The Cavendish knot is an identical heraldic knot. In shape, the Savoy knot is comparable to a figure eight.

When used outside heraldry (as a real knot), it is known as a figure-eight knot.

The Savoy knot can also be seen on the Alfa Romeo automobile badge (founded and manufactured in Milan, Italy) up to 1943.^[3]

• List of knots
• Order of the Most Holy Annunciation