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Journal of Knot Theory and Its Ramifications

Discipline	Mathematics
Language	English
Edited by	L. H. Kauffman
Publication details
History	1992-present
Publisher	World Scientific (Singapore)
Impact factor	0.363 (2016)
Standard abbreviations ISO 4 (alt) · Bluebook (alt) NLM (alt) · MathSciNet (alt )
ISO 4	J. Knot Theory Ramif.
MathSciNet	J. Knot Theory Ramifications
Indexing CODEN (alt · alt2) · JSTOR (alt) · LCCN (alt) MIAR · NLM (alt) · Scopus · W&L
ISSN	0218-2165 (print) 1793-6527 (web)
Links
Journal homepage

The Journal of Knot Theory and Its Ramifications was established in 1992 by Louis Kauffman and was the first journal purely devoted to knot theory. It is an interdisciplinary journal covering developments in knot theory, with emphasis on creating connections between with other branches of mathematics and the natural sciences. The journal is published by World Scientific.^[1]

According to the Journal Citation Reports, the journal has a 2020 impact factor of 0.379.

The journal is abstracted and indexed in:

• Science Citation Index
• ISI Alerting Services
• CompuMath Citation Index
• Current Contents/Physical, Chemical & Earth Sciences
• Mathematical Reviews
• Zentralblatt MATH

• History of knot theory

• Official website

In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984.^[1][2] Specifically, it is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polynomial in the variable ${\displaystyle t^{1/2}}$ with integer coefficients.^[3]

Suppose we have an oriented link ${\displaystyle L}$, given as a knot diagram. We will define the Jones polynomial ${\displaystyle V(L)}$ by using Louis Kauffman’s bracket polynomial, which we denote by ${\displaystyle \langle ~\rangle }$. Here the bracket polynomial is a Laurent polynomial in the variable ${\displaystyle A}$ with integer coefficients.

First, we define the auxiliary polynomial (also known as the normalized bracket polynomial)

${\displaystyle X(L)=(-A^{3})^{-w(L)}\langle L\rangle ,}$

where ${\displaystyle w(L)}$ denotes the writhe of ${\displaystyle L}$ in its given diagram. The writhe of a diagram is the number of positive crossings (${\displaystyle L_{+}}$ in the figure below) minus the number of negative crossings (${\displaystyle L_{-}}$). The writhe is not a knot invariant.

${\displaystyle X(L)}$ is a knot invariant since it is invariant under changes of the diagram of ${\displaystyle L}$ by the three Reidemeister moves. Invariance under type II and III Reidemeister moves follows from invariance of the bracket under those moves. The bracket polynomial is known to change by a factor of ${\displaystyle -A^{\pm 3}}$ under a type I Reidemeister move. The definition of the ${\displaystyle X}$ polynomial given above is designed to nullify this change, since the writhe changes appropriately by ${\displaystyle +1}$ or ${\displaystyle -1}$ under type I moves.

Now make the substitution ${\displaystyle A=t^{-1/4}}$ in ${\displaystyle X(L)}$ to get the Jones polynomial ${\displaystyle V(L)}$. This results in a Laurent polynomial with integer coefficients in the variable ${\displaystyle t^{1/2}}$.

Jones’ original formulation of his polynomial came from his study of operator algebras. In Jones’ approach, it resulted from a kind of “trace” of a particular braid representation into an algebra which originally arose while studying certain models, e.g. the Potts model, in statistical mechanics.

Let a link L be given. A theorem of Alexander states that it is the trace closure of a braid, say with n strands. Now define a representation ${\displaystyle \rho }$ of the braid group on n strands, B_n, into the Temperley–Lieb algebra ${\displaystyle \operatorname {TL} _{n}}$ with coefficients in ${\displaystyle \mathbb {Z} [A,A^{-1}]}$ and ${\displaystyle \delta =-A^{2}-A^{-2}}$. The standard braid generator ${\displaystyle \sigma _{i}}$ is sent to ${\displaystyle A\cdot e_{i}+A^{-1}\cdot 1}$, where ${\displaystyle 1,e_{1},\dots ,e_{n-1}}$ are the standard generators of the Temperley–Lieb algebra. It can be checked easily that this defines a representation.

Take the braid word ${\displaystyle \sigma }$ obtained previously from ${\displaystyle L}$ and compute ${\displaystyle \delta ^{n-1}\operatorname {tr} \rho (\sigma )}$ where ${\displaystyle \operatorname {tr} }$ is the Markov trace. This gives ${\displaystyle \langle L\rangle }$, where ${\displaystyle \langle }$ ${\displaystyle \rangle }$ is the bracket polynomial. This can be seen by considering, as Louis Kauffman did, the Temperley–Lieb algebra as a particular diagram algebra.

An advantage of this approach is that one can pick similar representations into other algebras, such as the R-matrix representations, leading to “generalized Jones invariants”.

The Jones polynomial is characterized by taking the value 1 on any diagram of the unknot and satisfies the following skein relation:

${\displaystyle (t^{1/2}-t^{-1/2})V(L_{0})=t^{-1}V(L_{+})-tV(L_{-})\,}$

where ${\displaystyle L_{+}}$, ${\displaystyle L_{-}}$, and ${\displaystyle L_{0}}$ are three oriented link diagrams that are identical except in one small region where they differ by the crossing changes or smoothing shown in the figure below:

The definition of the Jones polynomial by the bracket makes it simple to show that for a knot ${\displaystyle K}$, the Jones polynomial of its mirror image is given by substitution of ${\displaystyle t^{-1}}$ for ${\displaystyle t}$ in ${\displaystyle V(K)}$. Thus, an amphicheiral knot, a knot equivalent to its mirror image, has palindromic entries in its Jones polynomial. See the article on skein relation for an example of a computation using these relations.

Another remarkable property of this invariant states that the Jones polynomial of an alternating link is an alternating polynomial. This property was proved by Morwen Thistlethwaite^[5] in 1987. Another proof of this last property is due to Hernando Burgos-Soto, who also gave an extension of the property to tangles.^[6]

The Jones polynomial is not a complete invariant. There exist an infinite number of non-equivalent knots that have the same Jones polynomial. An example of two distinct knots having the same Jones polynomial can be found in the book by Murasugi.^[7]

For a positive integer ${\displaystyle N}$, the ${\displaystyle N}$-colored Jones polynomial ${\displaystyle V_{N}(L,t)}$ is a generalisation of the Jones polynomial. It is the Reshetikhin–Turaev invariant associated with the ${\displaystyle (N+1)}$-irreducible representation of the quantum group ${\displaystyle U_{q}({\mathfrak {sl}}_{2})}$. In this scheme, the Jones polynomial is the 1-colored Jones polynomial, the Reshetikhin-Turaev invariant associated to the standard representation (irreducible and two-dimensional) of ${\displaystyle U_{q}({\mathfrak {sl}}_{2})}$. One thinks of the strands of a link as being “colored” by a representation, hence the name.

More generally, given a link ${\displaystyle L}$ of ${\displaystyle k}$ components and representations ${\displaystyle V_{1},\ldots ,V_{k}}$ of ${\displaystyle U_{q}({\mathfrak {sl}}_{2})}$, the ${\displaystyle (V_{1},\ldots ,V_{k})}$-colored Jones polynomial ${\displaystyle V_{V_{1},\ldots ,V_{k}}(L,t)}$ is the Reshetikhin–Turaev invariant associated to ${\displaystyle V_{1},\ldots ,V_{k}}$ (here we assume the components are ordered). Given two representations ${\displaystyle V}$ and ${\displaystyle W}$, colored Jones polynomials satisfy the following two properties:^[8]

• ${\displaystyle V_{V\oplus W}(L,t)=V_{V}(L,t)+V_{W}(L,t)}$,
• ${\displaystyle V_{V\otimes W}(L,t)=V_{V,W}(L^{2},t)}$, where ${\displaystyle L^{2}}$ denotes the 2-cabling of ${\displaystyle L}$.

These properties are deduced from the fact that colored Jones polynomials are Reshetikhin-Turaev invariants.

Let ${\displaystyle K}$ be a knot. Recall that by viewing a diagram of ${\displaystyle K}$ as an element of the Temperley-Lieb algebra thanks to the Kauffman bracket, one recovers the Jones polynomial of ${\displaystyle K}$. Similarly, the ${\displaystyle N}$-colored Jones polynomial of ${\displaystyle K}$ can be given a combinatorial description using the Jones-Wenzl idempotents, as follows:

• consider the ${\displaystyle N}$-cabling ${\displaystyle K^{N}}$ of ${\displaystyle K}$;
• view it as an element of the Temperley-Lieb algebra;
• insert the Jones-Wenzl idempotents on some ${\displaystyle N}$ parallel strands.

The resulting element of ${\displaystyle \mathbb {Q} (t)}$ is the ${\displaystyle N}$-colored Jones polynomial. See appendix H of ^[9] for further details.

It is an open question whether there is a nontrivial knot with Jones polynomial equal to that of the unknot. It is known that there are nontrivial links with Jones polynomial equal to that of the corresponding unlinks by the work of Morwen Thistlethwaite.^[11] The simplest such example has 15 essential crossings and consists of a trefoil knot linked to a figure-eight knot. Every prime knot with up to 24 crossings, of which there are over three hundred million,^[12] is known to have a non-trivial Jones polynomial. This was determined by computing the Jones polynomial of several trillion knot diagrams, and verifying those diagrams with trivial Jones polynomials corresponded to trivial knots.^[13] It was shown by Kronheimer and Mrowka that there is no nontrivial knot with Khovanov homology equal to that of the unknot.^[14] While no nontrivial knot is known to have the same Jones polynomial as the unknot, two or more knots may have the same Jones polynomial and require other invariants to distinguish them. Examples include the Conway knot and the Kinoshita-Terasaka knot, with 11 crossings.^[15]

• Adams, Colin (2000-12-06). The Knot Book. American Mathematical Society. ISBN 0-8050-7380-9.
• Jones, Vaughan. “The Jones Polynomial” (PDF).
• Jones, Vaughan (1987). “Hecke algebra representations of braid groups and link polynomials”. Annals of Mathematics. 126 (2): 335–388. doi:10.2307/1971403. JSTOR 1971403.
• Kauffman, Louis H. (1987). “State models and the Jones polynomial”. Topology. 26 (3): 395–407. doi:10.1016/0040-9383(87)90009-7. (explains the definition by bracket polynomial and its relation to Jones’ formulation by braid representation)
• Lickorish, W. B. Raymond (1997). An introduction to knot theory. New York; Berlin; Heidelberg; Barcelona; Budapest; Hong Kong; London; Milan; Paris; Santa Clara; Singapore; Tokyo: Springer. p. 175. ISBN 978-0-387-98254-0.
• Thistlethwaite, Morwen (2001). “Links with trivial Jones polynomial”. Journal of Knot Theory and Its Ramifications. 10 (4): 641–643. doi:10.1142/S0218216501001050.
• Eliahou, Shalom; Kauffman, Louis H.; Thistlethwaite, Morwen B. (2003). “Infinite families of links with trivial Jones polynomial”. Topology. 42 (1): 155–169. doi:10.1016/S0040-9383(02)00012-5.
• Przytycki, Józef H. (1991). “Skein modules of 3-manifolds”. Bulletin of the Polish Academy of Sciences. 39 (1–2): 91–100. arXiv:math/0611797.

• “Jones-Conway polynomial”, Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Links with trivial Jones polynomial by Morwen Thistlethwaite
• “The Jones Polynomial“, The Knot Atlas.

In mathematics, especially in the area of topology known as knot theory, an invertible knot is a knot that can be continuously deformed to itself, but with its orientation reversed. A non-invertible knot is any knot which does not have this property. The invertibility of a knot is a knot invariant. An invertible link is the link equivalent of an invertible knot.

There are only five knot symmetry types, indicated by chirality and invertibility: fully chiral, reversible, positively amphichiral noninvertible, negatively amphichiral noninvertible, and fully amphichiral invertible.^[1]

It has long been known that most of the simple knots, such as the trefoil knot and the figure-eight knot are invertible. In 1962 Ralph Fox conjectured that some knots were non-invertible, but it was not proved that non-invertible knots exist until Hale Trotter discovered an infinite family of pretzel knots that were non-invertible in 1963.^[2] It is now known almost all knots are non-invertible.^[3]

All knots with crossing number of 7 or less are known to be invertible. No general method is known that can distinguish if a given knot is invertible.^[4] The problem can be translated into algebraic terms,^[5] but unfortunately there is no known algorithm to solve this algebraic problem.

If a knot is invertible and amphichiral, it is fully amphichiral. The simplest knot with this property is the figure eight knot. A chiral knot that is invertible is classified as a reversible knot.^[6]

The simplest example of a non-invertible knot is the knot 8₁₇ (Alexander-Briggs notation) or .2.2 (Conway notation). The pretzel knot 7, 5, 3 is non-invertible, as are all pretzel knots of the form (2p + 1), (2q + 1), (2r + 1), where p, q, and r are distinct integers, which is the infinite family proven to be non-invertible by Trotter.^[2]

• Chiral knot

• Jablan, Slavik & Sazdanovic, Radmila. Basic graph theory: Non-invertible knot and links Archived 2011-01-18 at the Wayback Machine, LinKnot.
• Explanation with a video, Nrich.Maths.org.

International Guild of Knot Tyers

Founded	17 April 1982; 44 years ago
Founder	25 Founding Members
Type	Educational non-profit
Focus	Knots and knotting techniques
Location	International
Region served	Global
Key people	Founders; Des Pawson and Geoffrey Budworth
Website	http://www.igkt.net/

The International Guild of Knot Tyers (or IGKT) is a worldwide association for people with an interest in knots and knot tying.

Officially established in 1982, the founding members were initially drawn together by the 1978 publication in The Times^[1] of an allegedly new knot, the Hunter’s bend.^[2] The idea for a knotting association of some kind grew from the contact between two people. Des Pawson was a retail manager for a large stationery firm based in Ipswich and a knot craftsman. Geoffrey Budworth was a Metropolitan Police Inspector and knotting consultant. Des first wrote to Geoff on 8 October 1978. They met before the month was over, and if it was not mentioned then the idea of contacting other knotting enthusiasts was raised by Des in a letter dated July, 1980, when he pressed for a suitable venue and suggested The Maritime Trust. Even then, 1981 went by without further development; and this is a source of regret to them both as it was the centenary of Clifford W. Ashley’s birth.

Knotting Matters is the quarterly news letter of the IGKT and is sent by post to all subscribed members.
The first issue was published in autumn 1982 and was 17 pages long and in black and white, edited by Geoffrey Budworth. The centennial was produced in September 2008, edited by Lindsey Philpott, and was professionally printed with colour covers and was 50 pages in length. Knotting Matters is made from Guild members’ submissions and other news from the guild.

The Guild dates from an inaugural meeting of 25 individuals aboard the Maritime Trust’s vessel RRS Discovery berthed in St. Katharine Docks in the lee of Tower Bridge on April, 17th. 1982. Those in attendance were Dr. Harry Asher, Roy E. Bail, C.G. Bellingham, Geoffrey Budworth, John Constable, Bernard J. Cutbush, Anne Devine, Ron W. Evans, Sid Evans, Eric Franklin, Frank Harris, John Hawes, Paul Herbert, Edward Hunter, Jill Jenner, Albert Kirby, Allan McDowall, Desmond Mandeville, Graham Mott, Des Pawson, Liz Pawson, Douglas Probert, W. Ettrick Thomson, Don Woods, and Quinton Winch.

Percy Blandford sent apologies for being unable to attend. Fred Browne of Boston, Massachusetts (MIT), Robert Chisnall of Kingston, Ontario, Canada, and Charles H. S. Thomason of Queensland, Australia all expressed a wish to be involved from the outset but due to distance were unable to attend the opening meeting. ^[7]

• Alaskan Branch
• East Anglian Branch (UK)
• French Branch
• German Branch
• Midlands Branch (UK)
• Northwest Branch (UK)
• Netherlands
• New Zealand
• NAB Mid-Atlantic Region
• NAB Knot In the Middle
• Pacific Americas Branch
• Solent Branch
• Surrey Branch
• Southern Ontario Knot Tyers Archived 12 April 2015 at the Wayback Machine
• Sweden
• West Country Knotters (UK)
• West Yorkshire Branch (UK)

In 2001, archaeological historian Mike Loades attempted a reconstruction of a British Iron Age chariot. He called upon IGKT member Richard Hopkins for his knowledge and experience of how to use the binding and lashing materials available at that time – rawhide, hemp, and flax – and described his contribution to the project as “invaluable”.^[8]

This involves tying six basic knots – reef knot, sheet bend, sheepshank, clove hitch, round turn and two half-hitches and bowline – against the clock. The authenticated world record is 8.1 seconds, set by Clinton R. Bailey, Sr. in 1977.^[9] IGKT members have discussed proposals for formal rules to govern future attempts on this record.^[10][11]

In 2018, the IGKT-Solent Branch promoted the idea of making the 18th of December World Knot Tying Day to celebrate and remember the author Clifford W. Ashley, who wrote and illustrated The Ashley Book of Knots. The date was selected to coincide with Ashley’s birthday (1881). Participants were asked to tie their favorite knot and also learn a new knot. They were also encouraged to teach someone how to tie a knot. Even teaching someone to tie their shoelaces was sufficient. When the knots were tied, participants were encouraged to post a photo of their knot on their favorite social media site with the hashtag #WorldKnotTyingDay.^[12] In 2020, the IGKT shifted the day of the celebration to September 18. This date coincides with the day Ashley died (1947).

• List of knots
• Category:Knots

• Official website of the International Guild of Knot Tyers
• Site officiel de l’IGKT France
• Video documentary on the history of the IGKT
• Six Knot Challenge on YouTube Two young Canadian Scouts, coached by members of the Southern Ontario Knot Tyers, tie the six basic Scout knots in record videorecorded time

In the visual arts, interlace is a decorative element found in medieval art. In interlace, bands or portions of other motifs are looped, braided, and knotted in complex geometric patterns, often to fill a space. Interlacing is common in the Migration period art of Northern Europe, in the early medieval Insular art of Britain and Ireland, and Norse art of the Early Middle Ages, and in Islamic art.

Intricate braided and interlaced patterns, called plaits in British usage, first appeared in late Roman art in various parts of Europe, in mosaic floors and other media. Coptic manuscripts and textiles of 5th- and 6th-century Christian Egypt are decorated with broad-strand ribbon interlace ornament bearing a “striking resemblance” to the earliest types of knotwork found in the Insular art manuscripts of Ireland and the British Isles.^[1]

• 
  Roman interlace on a floor from a private house in Gerasa, Jordan, 2nd century, mosaic, Pergamon Museum, Berlin
• 
  Roman interlaces on a mosaic floor, Villa Romana del Casale, near Piazza Armerina, Italy, unknown architect, early 4th century
• 
  Roman interlaces on a floor, 4th-6th centuries, mosaic, Constanța History and Archaeology Museum, Constanța, Romania
• 
  Insular belt buckle from Sutton Hoo, 580–620, gold and niello, British Museum, London
• 
  Insular animal and knot interlace in the Lindisfarne Gospels, early 8th century, ink and pigments on paper, British Library, London
• 
  Page from the Book of Dimma with simple Insular interlace borders, 8th century, illuminated manuscript, Library of Trinity College Dublin, Dublin, Ireland
• 
  Cross decorated with Insular interlaces, part of the Aberlemno Sculptured Stones, unknown sculptor, c.800, sandstone, Aberlemo, Scotland, UK
• 
  Detail of decorated Insular initial “T” with ribbon interlace filling and interlaced animal motif, Book of Kells, c.800, illuminated manuscript, Library of Trinity College Dublin
• 
  Inscription of Stephen Držislav of Croatia, an example of the three-strand Croatian interlace, 10th century, stone, Croatian Institute of History, Zagreb, Croatia
• 
  Byzantine interlaces on a slab, 11th century, marble, Byzantine and Christian Museum, Athens
• 
  Romanesque interlaces on the only surviving original wooden door of St. Maria im Kapitol, Cologne, Germany, unknown architect or sculptor, c.1065
• 
  Uppland Runic Inscription 1014 with Viking interlaced animal, Uppland, Sweden, attributed to the runemaster Öpir, late 11th or early 12th century
• 
  Byzantine interlaces on the cover of the Melisende Psalter, 1131-1143, ivory, British Library ^[7]
• 
  Romanesque interlace on an initial “inhabited” with figures on a page of the Leiden Saint Louis Psalter, 1190-1200, ink and painting on parchment, Leiden University Library, Leiden, Netherlands
• 
  Islamic interlaces on a carpet page from the Ibn al-Bawwab Qur’an, by Ibn al-Bawwab, 11th century, ink and painting on paper, Chester Beatty Library, Dublin
• 
  Folio from a manuscript of the Qur’an with Islamic interlaced border, 1182, ink and painting on parchment, Istanbul University Library, Istanbul, Turkey
• 
  Islamic interlace on a tile from an architectural frieze, 1380-1420, glazed earthenware, Victoria and Albert Museum, London
• 
  Renaissance interlaces on a page from a codex with a portrait of Matthias Corvinus, by Giovanni Ambrogio de Predis, 16th century, illuminated manuscript, unknown location
• 
  Neoclassical interlace on a wall in the Neues Museum, Berlin, by Friedrich August Stüler, 1843-1855
• 
  Art Nouveau interlaces on the shafts of the entrance portal columns of the Watts Cemetery Chapel, village cemetery of Compton, Guildford, England, inspired by Insular art, by Mary Fraser Tytler and George Redmayne, 1898
• 
  Art Nouveau interlace on a stove in the George Severeanu Museum, Bucharest, Romania, unknown architect, c.1900
• 
  Romanian Revival interlaces on the Vlahuți-Slătineanu Grave, Bellu Cemetery, Bucharest, by Grigore Cerkez, 1913
• 
  Romanian Revival interlaces on a corbel of Strada Vasile Lascăr no. 76, Bucharest, by Ștefan Ciocârlan, 1925^[8]
• 
  Romanian Revival interlaces on the Dimitrie Vișinescu Family Grave, Bellu Cemetery, unknown architect, c.1930

• Bain, George (1973). Celtic Art: The Methods of Construction. Dover Publications, Inc. ISBN 0-486-22923-8.
• Bishop, Morris (2001). The Middle Ages. Mariner Books. ISBN 0-618-05703-X.
• Graham-Campbell, James (1980). The Viking World. Ticknor & Fields. ISBN 0-89919-005-7.
• Mitchell, G. Frank, Peter Harbison, Liam de Paor, Máire de Paor, and Roger A. Stalley (1977). Treasures of Irish Art, 1500 B.C. to 1500 A.D. : From the Collections of the National Museum of Ireland, Royal Irish Academy, & Trinity College, Dublin. Metropolitan Museum of Art & Alfred A. Knopf, New York. ISBN 0-394-42807-2.{{cite book}}: CS1 maint: multiple names: authors list (link)

• Illustrated article by Peter Hubert on the origins of interlace sculpture. Archived 2016-11-18 at the Wayback Machine

In the mathematical field of knot theory, the hyperbolic volume of a hyperbolic link is the volume of the link’s complement with respect to its complete hyperbolic metric. The volume is necessarily a finite real number, and is a topological invariant of the link.^[1] As a link invariant, it was first studied by William Thurston in connection with his geometrization conjecture.^[2]

A hyperbolic link is a link in the 3-sphere whose complement (the space formed by removing the link from the 3-sphere) can be given a complete Riemannian metric of constant negative curvature, giving it the structure of a hyperbolic 3-manifold, a quotient of hyperbolic space by a group acting freely and discontinuously on it. The components of the link will become cusps of the 3-manifold, and the manifold itself will have finite volume. By Mostow rigidity, when a link complement has a hyperbolic structure, this structure is uniquely determined, and any geometric invariants of the structure are also topological invariants of the link. In particular, the hyperbolic volume of the complement is a knot invariant. In order to make it well-defined for all knots or links, the hyperbolic volume of a non-hyperbolic knot or link is often defined to be zero.

There are only finitely many hyperbolic knots for any given volume.^[2] A mutation of a hyperbolic knot will have the same volume,^[3] so it is possible to concoct examples with equal volumes; indeed, there are arbitrarily large finite sets of distinct knots with equal volumes.^[2]
In practice, hyperbolic volume has proven very effective in distinguishing knots, utilized in some of the extensive efforts at knot tabulation. Jeffrey Weeks’s computer program SnapPea is the ubiquitous tool used to compute hyperbolic volume of a link.^[1]

Knot/link	Volume	Reference
Figure-eight knot	${\displaystyle \textstyle -6\int _{0}^{\pi /3}{\log {|2\sin \theta |}d\theta }=2.02988…}$	[4]
Three-twist knot	2.82812
Stevedore knot	3.16396
62 knot	4.40083
Endless knot	5.13794
Perko pair	5.63877
63 knot	5.69302
Borromean rings	${\displaystyle \textstyle -16\int _{0}^{\pi /4}{\log {|2\sin \theta |}d\theta }=7.32772…}$	[4]

More generally, the hyperbolic volume may be defined for any hyperbolic 3-manifold. The Weeks manifold has the smallest possible volume of any closed manifold (a manifold that, unlike link complements, has no cusps); its volume is approximately 0.9427.^[5]

Thurston and Jørgensen proved that the set of real numbers that are hyperbolic volumes of orientable 3-manifolds is well-ordered, with order type ω^ω. For each 0 ≤ k < l, each volume of a manifold with exactly l cusps is a limit point of a sequence of manifolds with k cusps, and every convergent sequence of volumes is obtainable by Dehn surgeries on a manifold at the limiting volume; volumes of closed manifolds are exactly the isolated points.^[6] The smallest limit point in this set of volumes is given by the knot complement of the figure-eight knot,^[7] and the smallest limit point of limit points is given by the complement of the Whitehead link.^[8]

• “Hyperbolic Volume“, The Knot Atlas.

Hunter’s bend
Names	Hunter’s bend, Rigger’s bend
Category	Bend
Related	Alpine butterfly bend, Zeppelin bend, Zeppelin loop, Ashley’s bend
Typical use	easily-untied bend for rigging and mooring
ABoK	#1425A

Hunter’s bend (or rigger’s bend) is a knot used to join two lines. It consists of interlocking overhand knots, and can jam under moderate strain. It is similar to the Zeppelin bend.

When assessed against other bends in stress tests using paracord, it was found to be “not as strong as the blood knot, similar to the reverse figure of eight and stronger than the fisherman’s bend, sheet bend or reef knot“.^[1]

In October 1978, an article in The Times presented it as a newly invented knot credited to Dr. Edward Hunter.^[2] He had used it for years to tie broken shoelaces before discovering its originality through a friend in the 1970s. When it appeared on the front page, it led to much publicity for the knot and also to the formation of the International Guild of Knot Tyers.^[3]

It was later pointed out by Amory Bloch Lovins that the knot had already been presented in Knots for Mountaineering by Phil D. Smith in the 1950s.^[4] The tying of the bend was described as a modification to the alpine butterfly bend.^[5] Smith had devised the knot in 1943 while working on the San Francisco waterfront and had called it simply a “rigger’s bend”.^[1][3]

Although not documented in the original 1944 print of The Ashley Book of Knots, it was later added in 1979 as entry #1425A.^[1]

• List of knots
• Zeppelin bend
• List of bend knots

• Alternative steps to tie a Hunter’s Bend

The Hungerford or Hastings knot is a heraldic knot used as an heraldic badge in English heraldry by the Hungerford and Hastings families. The binding together of a Hungerford sickle and a Peverell garb (wheatsheaf) with the Hungerford knot commemorates the marriage between the Hungerfords and the Peverells in the early 15th century.^[1]

The Hoxton knot,^[1] Chelsea knot,^[2] French loop,^[3] Parisian scarf knot^[4] or Snug Tug^[5] is a method of arranging a scarf about the neck. The scarf is doubled back and placed around the neck. The tails of the scarf are then pulled through the U-bend of the doubling to secure them, as with a cow hitch or lark’s head.

The knot is popular with stylish men like David Beckham who frequent fashionable districts of London such as Hoxton and Chelsea.^[1] The style is also commonly used by outside broadcasters from the BBC as it is warm and tidy.^[3] It may be controversial though, as some commentators opine that knotting a scarf is less manly than just draping it around the neck or throwing the ends casually over the shoulder.^[6]

• Snood – the snug, tubular comforter which generated similar controversy when worn by football players.

Knots have been used for basic purposes such as recording information, fastening and tying objects together, for thousands of years. The early significant stimulus in knot theory would arrive later with Sir William Thomson (Lord Kelvin) and his vortex theory of the atom.

• Silver, Dan, Scottish physics and knot theory’s odd origins Archived 2015-09-24 at the Wayback Machine (expanded version of Silver, “Knot theory’s odd origins,” American Scientist, 94, No. 2, 158–165)
• J.C. Turner & P. van de Griend, editors (1995) History and Science of Knots, World Scientific.
• Bar-Natan, Dror (1995), “On the Vassiliev knot invariants”, Topology, 34 (2): 423–472, doi:10.1016/0040-9383(95)93237-2
• Kontsevich, M. (1993). “Vassiliev’s knot invariants”. I. M. Gelfand Seminar. ADVSOV. Vol. 16. pp. 137–150. doi:10.1090/advsov/016.2/04. ISBN 978-0-8218-4117-4.
• Turaev, Vladimir G. (2016) [1994]. Quantum Invariants of Knots and 3-Manifolds. doi:10.1515/9783110435221. ISBN 978-3-11-043522-1. S2CID 118682559.
• Witten, Edward (1989), “Quantum field theory and the Jones polynomial”, Comm. Math. Phys., 121 (3): 351–399, Bibcode:1989CMaPh.121..351W, doi:10.1007/BF01217730, S2CID 14951363

• Thomson, Sir William (Lord Kelvin), On Vortex Atoms, Proceedings of the Royal Society of Edinburgh, Vol. VI, 1867, pp. 94–105.
• Silliman, Robert H., William Thomson: Smoke Rings and Nineteenth-Century Atomism, Isis, Vol. 54, No. 4. (Dec., 1963), pp. 461–474. JSTOR link
• Movie Archived 2015-09-24 at the Wayback Machine of a modern recreation of Tait’s smoke ring experiment
• The history of knots