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Knot theory
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File:Tabela de nÃ³s matemÃ¡ticos 01, crop.jpg|thumb|Examples of different knots including the trivial knot (top left) and (below it) the trefoil knottrefoil knot(File:TrefoilKnot 01.svg|thumb|A knot diagram of the trefoil knot, the simplest non-trivial knot)In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined together so that it cannot be undone, the simplest knot being a ring (or "unknot"). In mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3 (in topology, a circle isn't bound to the classical geometric concept, but to all of its homeomorphisms). Two mathematical knots are equivalent if one can be transformed into the other via a deformation of R3 upon itself (known as an ambient isotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself.Knots can be described in various ways. Given a method of description, however, there may be more than one description that represents the same knot. For example, a common method of describing a knot is a planar diagram called a knot diagram. Any given knot can be drawn in many different ways using a knot diagram. Therefore, a fundamental problem in knot theory is determining when two descriptions represent the same knot.A complete algorithmic solution to this problem exists, which has unknown complexity. In practice, knots are often distinguished by using a knot invariant, a "quantity" which is the same when computed from different descriptions of a knot. Important invariants include knot polynomials, knot groups, and hyperbolic invariants.The original motivation for the founders of knot theory was to create a table of knots and links, which are knots of several components entangled with each other. More than six billion knots and links have been tabulated since the beginnings of knot theory in the 19th century.To gain further insight, mathematicians have generalized the knot concept in several ways. Knots can be considered in other three-dimensional spaces and objects other than circles can be used; see knot (mathematics). Higher-dimensional knots are n-dimensional spheres in m-dimensional Euclidean space.

## History

File:KellsFol034rXRhoDet3.jpeg|thumb|upright=.85|Intricate Celtic knotwork in the 1200-year-old Book of KellsBook of KellsArchaeologists have discovered that knot tying dates back to prehistoric times. Besides their uses such as recording information and tying objects together, knots have interested humans for their aesthetics and spiritual symbolism. Knots appear in various forms of Chinese artwork dating from several centuries BC (see Chinese knotting). The endless knot appears in Tibetan Buddhism, while the Borromean rings have made repeated appearances in different cultures, often representing strength in unity. The Celtic monks who created the Book of Kells lavished entire pages with intricate Celtic knotwork.File:Peter Tait.jpg|thumb|left|upright|The first knot tabulator, Peter Guthrie TaitPeter Guthrie TaitA mathematical theory of knots was first developed in 1771 by Alexandre-ThÃ©ophile Vandermonde who explicitly noted the importance of topological features when discussing the properties of knots related to the geometry of position. Mathematical studies of knots began in the 19th century with Carl Friedrich Gauss, who defined the linking integral {{Harv|Silver|2006}}. In the 1860s, Lord Kelvin's theory that atoms were knots in the aether led to Peter Guthrie Tait's creation of the first knot tables for complete classification. Tait, in 1885, published a table of knots with up to ten crossings, and what came to be known as the Tait conjectures. This record motivated the early knot theorists, but knot theory eventually became part of the emerging subject of topology.These topologists in the early part of the 20th centuryâ€”Max Dehn, J. W. Alexander, and othersâ€”studied knots from the point of view of the knot group and invariants from homology theory such as the Alexander polynomial. This would be the main approach to knot theory until a series of breakthroughs transformed the subject.In the late 1970s, William Thurston introduced hyperbolic geometry into the study of knots with the hyperbolization theorem. Many knots were shown to be hyperbolic knots, enabling the use of geometry in defining new, powerful knot invariants. The discovery of the Jones polynomial by Vaughan Jones in 1984 {{Harv|Sossinsky|2002|pp=71â€“89}}, and subsequent contributions from Edward Witten, Maxim Kontsevich, and others, revealed deep connections between knot theory and mathematical methods in statistical mechanics and quantum field theory. A plethora of knot invariants have been invented since then, utilizing sophisticated tools such as quantum groups and Floer homology.In the last several decades of the 20th century, scientists became interested in studying physical knots in order to understand knotting phenomena in DNA and other polymers. Knot theory can be used to determine if a molecule is chiral (has a "handedness") or not {{Harv|Simon|1986}}. Tangles, strings with both ends fixed in place, have been effectively used in studying the action of topoisomerase on DNA {{Harv|Flapan|2000}}. Knot theory may be crucial in the construction of quantum computers, through the model of topological quantum computation {{Harv|Collins|2006}}.

## Knot equivalence

{{multiple image
| align=right
| total_width=370
| image1=unknots.svg
| width1=289 | height1=240
| image2=Knot Unfolding.gif
| width2=240 | height2=240
| footer=On the left, the unknot, and a knot equivalent to it. It can be more difficult to determine whether complex knots, such as the one on the right, are equivalent to the unknot.
}}A knot is created by beginning with a one-dimensional line segment, wrapping it around itself arbitrarily, and then fusing its two free ends together to form a closed loop {{Harv|Adams|2004}} {{Harv|Sossinsky|2002}}. Simply, we can say a knot K is a "simple closed curve" or "(closed) Jordan curve" (see Curve) â€” that is: a "nearly" injective and continuous function Kcolon[0,1]to mathbb{R}^3, with the only "non-injectivity" being K(0)=K(1). Topologists consider knots and other entanglements such as links and braids to be equivalent if the knot can be pushed about smoothly, without intersecting itself, to coincide with another knot.The idea of knot equivalence is to give a precise definition of when two knots should be considered the same even when positioned quite differently in space. A formal mathematical definition is that two knots K_1, K_2 are equivalent if there is an orientation-preserving homeomorphism hcolonR^3toR^3 with h(K_1)=K_2.Another way to define knot equivalence is that two knots are equivalent when there is a continuous family of homeomorphisms {ht: R3 â†’ R3 for 0 â‰¤ t â‰¤ 1} of space onto itself, such that the last one of them carries the first knot onto the second knot. (More formally: Two knots K1 and K2 are equivalent if there exists a continuous mapping H: R3Ã— [0,1] â†’ R3 such that a) for each t âˆˆ [0,1] the mapping taking x âˆˆ R3 to H(x,t) âˆˆ R3 is a homeomorphism of R3 onto itself; b) H(x, 0) = x for all x âˆˆ R3; and c) H(K1,1) = K2. Such a function H is known as an ambient isotopy.)These two notions of knot equivalence agree exactly about which knots are equivalent: Two knots that are equivalent under the orientation-preserving homeomorphism definition are also equivalent under the ambient isotopy definition, because any orientation-preserving homeomorphisms of R3 to itself is the final stage of an ambient isotopy starting from the identity. Conversely, two knots equivalent under the ambient isotopy definition are also equivalent under the orientation-preserving homeomorphism definition, because the t=1 (final) stage of the ambient isotopy must be an orientation-preserving homeomorphism carrying one knot to the other.The basic problem of knot theory, the recognition problem, is determining the equivalence of two knots. Algorithms exist to solve this problem, with the first given by Wolfgang Haken in the late 1960s {{Harv|Hass|1998}}. Nonetheless, these algorithms can be extremely time-consuming, and a major issue in the theory is to understand how hard this problem really is {{Harv|Hass|1998}}. The special case of recognizing the unknot, called the unknotting problem, is of particular interest {{Harv|Hoste|2005}}.

## Knot diagrams

A useful way to visualise and manipulate knots is to project the knot onto a planeâ€”think of the knot casting a shadow on the wall. A small change in the direction of projection will ensure that it is one-to-one except at the double points, called crossings, where the "shadow" of the knot crosses itself once transversely {{Harv|Rolfsen|1976}}. At each crossing, to be able to recreate the original knot, the over-strand must be distinguished from the under-strand. This is often done by creating a break in the strand going underneath. The resulting diagram is an immersed plane curve with the additional data of which strand is over and which is under at each crossing. (These diagrams are called knot diagrams when they represent a knot and link diagrams when they represent a link.) Analogously, knotted surfaces in 4-space can be related to immersed surfaces in 3-space.A reduced diagram is a knot diagram in which there are no reducible crossings (also nugatory or removable crossings), or in which all of the reducible crossings have been removed.{{harv|Weisstein||loc=ReducedKnotDiagram}}{{harv|Weisstein||loc=ReducibleCrossing}}

### Reidemeister moves

In 1927, working with this diagrammatic form of knots, J. W. Alexander and Garland Baird Briggs, and independently Kurt Reidemeister, demonstrated that two knot diagrams belonging to the same knot can be related by a sequence of three kinds of moves on the diagram, shown below. These operations, now called the Reidemeister moves, are:{{Ordered list|list-style-type=upper-Roman|Twist and untwist in either direction.|Move one strand completely over another.|Move a strand completely over or under a crossing.}}{| align="center" style="text-align:center"|+ Reidemeister moves style="padding:1em"
130px) (File:Frame left.png) (File:Reidemeister move 2.png|210px)
! Type I !! Type II
(File:Reidemeister move 3.png|360px)
! colspan="2" | Type III
The proof that diagrams of equivalent knots are connected by Reidemeister moves relies on an analysis of what happens under the planar projection of the movement taking one knot to another. The movement can be arranged so that almost all of the time the projection will be a knot diagram, except at finitely many times when an "event" or "catastrophe" occurs, such as when more than two strands cross at a point or multiple strands become tangent at a point. A close inspection will show that complicated events can be eliminated, leaving only the simplest events: (1) a "kink" forming or being straightened out; (2) two strands becoming tangent at a point and passing through; and (3) three strands crossing at a point. These are precisely the Reidemeister moves {{Harv|Sossinsky|2002| loc=ch. 3}} {{Harv|Lickorish|1997| loc=ch. 1}}.

## Knot invariants

A knot invariant is a "quantity" that is the same for equivalent knots {{Harv|Adams|2004}} {{Harv|Lickorish|1997}} {{Harv|Rolfsen|1976}}. For example, if the invariant is computed from a knot diagram, it should give the same value for two knot diagrams representing equivalent knots. An invariant may take the same value on two different knots, so by itself may be incapable of distinguishing all knots. An elementary invariant is tricolorability."Classical" knot invariants include the knot group, which is the fundamental group of the knot complement, and the Alexander polynomial, which can be computed from the Alexander invariant, a module constructed from the infinite cyclic cover of the knot complement {{Harv|Lickorish|1997}}{{Harv|Rolfsen|1976}}. In the late 20th century, invariants such as "quantum" knot polynomials, Vassiliev invariants and hyperbolic invariants were discovered. These aforementioned invariants are only the tip of the iceberg of modern knot theory.

### Knot polynomials

A knot polynomial is a knot invariant that is a polynomial. Well-known examples include the Jones and Alexander polynomials. A variant of the Alexander polynomial, the Alexanderâ€“Conway polynomial, is a polynomial in the variable z with integer coefficients {{Harv|Lickorish|1997}}.The Alexanderâ€“Conway polynomial is actually defined in terms of links, which consist of one or more knots entangled with each other. The concepts explained above for knots, e.g. diagrams and Reidemeister moves, also hold for links.Consider an oriented link diagram, i.e. one in which every component of the link has a preferred direction indicated by an arrow. For a given crossing of the diagram, let L_+, L_-, L_0 be the oriented link diagrams resulting from changing the diagram as indicated in the figure: (File:Skein (HOMFLY).svg|200px|center)The original diagram might be either L_+ or L_-, depending on the chosen crossing's configuration. Then the Alexanderâ€“Conway polynomial, C(z), is recursively defined according to the rules:
• C(O) = 1 (where O is any diagram of the unknot)
• C(L_+) = C(L_-) + z C(L_0).
The second rule is what is often referred to as a skein relation. To check that these rules give an invariant of an oriented link, one should determine that the polynomial does not change under the three Reidemeister moves. Many important knot polynomials can be defined in this way.The following is an example of a typical computation using a skein relation. It computes the Alexanderâ€“Conway polynomial of the trefoil knot. The yellow patches indicate where the relation is applied.
C((File:skein-relation-trefoil-plus-sm.png)) = C((File:skein-relation-trefoil-minus-sm.png)) + z C((File:skein-relation-trefoil-zero-sm.png))
gives the unknot and the Hopf link. Applying the relation to the Hopf link where indicated,
gives a link deformable to one with 0 crossings (it is actually the unlink of two components) and an unknot. The unlink takes a bit of sneakiness:
which implies that C(unlink of two components) = 0, since the first two polynomials are of the unknot and thus equal.Putting all this together will show:
C(trefoil) = 1 + z(0 + z) = 1 + z2.
Since the Alexanderâ€“Conway polynomial is a knot invariant, this shows that the trefoil is not equivalent to the unknot. So the trefoil really is "knotted".Image:Trefoil knot left.svg|The left-handed trefoil knot.Image:TrefoilKnot_01.svg|The right-handed trefoil knot.Actually, there are two trefoil knots, called the right and left-handed trefoils, which are mirror images of each other (take a diagram of the trefoil given above and change each crossing to the other way to get the mirror image). These are not equivalent to each other, meaning that they are not amphicheiral. This was shown by Max Dehn, before the invention of knot polynomials, using group theoretical methods {{Harv|Dehn|1914}}. But the Alexanderâ€“Conway polynomial of each kind of trefoil will be the same, as can be seen by going through the computation above with the mirror image. The Jones polynomial can in fact distinguish between the left- and right-handed trefoil knots {{Harv|Lickorish|1997}}.

### Hyperbolic invariants

William Thurston proved many knots are hyperbolic knots, meaning that the knot complement (i.e., the set of points of 3-space not on the knot) admits a geometric structure, in particular that of hyperbolic geometry. The hyperbolic structure depends only on the knot so any quantity computed from the hyperbolic structure is then a knot invariant {{Harv|Adams|2004}}.{{multiple image
| align = right
| total_width = 320
| image1 = BorromeanRings.svg
| width1 = 626 | height1 = 600
| caption1 = The Borromean rings are a link with the property that removing one ring unlinks the others.
| image2 = SnapPea-horocusp_view.png
| width2 = 560 | height2 = 416
| caption2 = SnapPea's cusp view: the Borromean rings complement from the perspective of an inhabitant living near the red component.

## Higher dimensions

A knot in three dimensions can be untied when placed in four-dimensional space. This is done by changing crossings. Suppose one strand is behind another as seen from a chosen point. Lift it into the fourth dimension, so there is no obstacle (the front strand having no component there); then slide it forward, and drop it back, now in front. Analogies for the plane would be lifting a string up off the surface, or removing a dot from inside a circle.In fact, in four dimensions, any non-intersecting closed loop of one-dimensional string is equivalent to an unknot. First "push" the loop into a three-dimensional subspace, which is always possible, though technical to explain.

### Knotting spheres of higher dimension

Since a knot can be considered topologically a 1-dimensional sphere, the next generalization is to consider a two-dimensional sphere (S2) embedded in 4-dimensional Euclidean space (R4). Such an embedding is knotted if there is no homeomorphism of R4 onto itself taking the embedded 2-sphere to the standard "round" embedding of the 2-sphere. Suspended knots and spun knots are two typical families of such 2-sphere knots.The mathematical technique called "general position" implies that for a given n-sphere in m-dimensional Euclidean space, if m is large enough (depending on n), the sphere should be unknotted. In general, piecewise-linear n-spheres form knots only in (n + 2)-dimensional space {{Harv|Zeeman|1963}}, although this is no longer a requirement for smoothly knotted spheres. In fact, there are smoothly knotted (4k − 1)-spheres in 6k-dimensional space, e.g. there is a smoothly knotted 3-sphere in R6 {{Harv|Haefliger|1962}}{{Harv|Levine|1965}}. Thus the codimension of a smooth knot can be arbitrarily large when not fixing the dimension of the knotted sphere; however, any smooth k-sphere embedded in Rn with 2n − 3k − 3 > 0 is unknotted. The notion of a knot has further generalisations in mathematics, see: knot (mathematics), isotopy classification of embeddings.Every knot in Sn is the link of a real-algebraic set with isolated singularity in Rn+1 {{Harv|Akbulut|King|1981}}.An n-knot is a single Sn embedded in Rm. An n-link is k-copies of Sn embedded in Rm, where k is a natural number. Both the m = n + 2 case and the m > n + 2 case are researched well.The n > 1 case has different futures from the n = 1 case and is an exciting field.{{citation |first=J. |last=Levine |first2=K |last2=Orr |chapter=A survey of applications of surgery to knot and link theory |citeseerx = 10.1.1.64.4359 |title=Surveys on Surgery Theory: Papers Dedicated to C.T.C. Wall |volume=1 |series=Annals of mathematics studies |publisher=Princeton University Press |year=2000 |isbn=978-0691049380 }} â€” An introductory article to high dimensional knots and links for the advanced readers{{citation |first=Eiji |last=Ogasa |arxiv=1304.6053 |title=Introduction to high dimensional knots |bibcode=2013arXiv1304.6053O |year=2013 }} â€” An introductory article to high dimensional knots and links for the beginners

(File:Sum of knots3.svg|thumb|Adding two knots)Two knots can be added by cutting both knots and joining the pairs of ends. The operation is called the knot sum, or sometimes the connected sum or composition of two knots. This can be formally defined as follows {{Harv|Adams|2004}}: consider a planar projection of each knot and suppose these projections are disjoint. Find a rectangle in the plane where one pair of opposite sides are arcs along each knot while the rest of the rectangle is disjoint from the knots. Form a new knot by deleting the first pair of opposite sides and adjoining the other pair of opposite sides. The resulting knot is a sum of the original knots. Depending on how this is done, two different knots (but no more) may result. This ambiguity in the sum can be eliminated regarding the knots as oriented, i.e. having a preferred direction of travel along the knot, and requiring the arcs of the knots in the sum are oriented consistently with the oriented boundary of the rectangle.The knot sum of oriented knots is commutative and associative. A knot is prime if it is non-trivial and cannot be written as the knot sum of two non-trivial knots. A knot that can be written as such a sum is composite. There is a prime decomposition for knots, analogous to prime and composite numbers {{Harv|Schubert|1949}}. For oriented knots, this decomposition is also unique. Higher-dimensional knots can also be added but there are some differences. While you cannot form the unknot in three dimensions by adding two non-trivial knots, you can in higher dimensions, at least when one considers smooth knots in codimension at least 3.

## Tabulating knots

### Alexanderâ€“Briggs notation

This is the most traditional notation, due to the 1927 paper of James W. Alexander and Garland B. Briggs and later extended by Dale Rolfsen in his knot table (see image above and List of prime knots). The notation simply organizes knots by their crossing number. One writes the crossing number with a subscript to denote its order amongst all knots with that crossing number. This order is arbitrary and so has no special significance (though in each number of crossings the twist knot comes after the torus knot). Links are written by the crossing number with a superscript to denote the number of components and a subscript to denote its order within the links with the same number of components and crossings. Thus the trefoil knot is notated 31 and the Hopf link is 2{{sup sub|2|1}}.

### Dowker notation

(File:Dowker-notation-example.svg|thumb|upright=.85|A knot diagram with crossings labelled for a Dowker sequence)The Dowker notation, also called the Dowkerâ€“Thistlethwaite notation or code, for a knot is a finite sequence of even integers. The numbers are generated by following the knot and marking the crossings with consecutive integers. Since each crossing is visited twice, this creates a pairing of even integers with odd integers. An appropriate sign is given to indicate over and undercrossing. For example, in this figure the knot diagram has crossings labelled with the pairs (1,6) (3,−12) (5,2) (7,8) (9,−4) and (11,−10). The Dowker notation for this labelling is the sequence: 6, −12, 2, 8, −4, −10. A knot diagram has more than one possible Dowker notation, and there is a well-understood ambiguity when reconstructing a knot from a Dowker notation.

### Gauss code

Gauss code, similar to Dowker notation, represents a knot with a sequence of integers. However, rather than every crossing being represented by two different numbers, crossings are labeled with only one number. When the crossing is an overcrossing, a positive number is listed. At an undercrossing, a negative number.For example, the trefoil knot in Gauss code can be given as: 1,âˆ’2,3,âˆ’1,2,âˆ’3Gauss code is limited in its ability to identify knots by a few problems. The starting point on the knot at which to begin tracing the crossings is arbitrary, and there is no way to determine which direction to trace in. Also, Gauss code is unable to indicate the handedness of each crossing, which is necessary to identify a knot versus its mirror. For example, the Gauss code for the trefoil knot does not specify if it is the right-handed or left-handed trefoil.This last issue is often solved with extended Gauss code. In this modification, the positive/negative sign on the second instance of every number is chosen to represent the handedness of that crossing, rather than the over/under sign of the crossing, which is made clear in the first instance of the number. A right-handed crossing is given a positive number, and a left handed crossing is given a negative number.

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• WEB,weblink Reducible Crossing, Weisstein, Eric W., MathWorld, Wolfram, 8 May 2013, harv,
• {{Citation|last=Witten|first=Edward|author-link=Edward Witten|title=Quantum field theory and the Jones polynomial|journal=Comm. Math. Phys.|volume=121|year=1989|pages=351â€“399|issue=3|doi=10.1007/BF01217730|bibcode=1989CMaPh.121..351W|ref=harv}}
• {{Citation|first=E. C.|last=Zeeman|author-link=E. C. Zeeman|title=Unknotting combinatorial balls|journal=Annals of Mathematics |series=Second Series|year=1963|volume=78|pages=501â€“526|issue=3|jstor=1970538|ref=harv|doi=10.2307/1970538}}
{{reflist}}

### Introductory textbooks

There are a number of introductions to knot theory. A classical introduction for graduate students or advanced undergraduates is {{Harv|Rolfsen|1976}}. Other good texts from the references are {{Harv|Adams|2001}} and {{Harv|Lickorish|1997}}. Adams is informal and accessible for the most part to high schoolers. Lickorish is a rigorous introduction for graduate students, covering a nice mix of classical and modern topics.
• {{citation |first=Gerhard |last=Burde |authorlink=Gerhard Burde |author2-link=Heiner Zieschang |first2=Heiner |last2=Zieschang |title=Knots |publisher=Walter de Gruyter |series=De Gruyter Studies in Mathematics |volume=5 |year=1985 |isbn=978-3-11-008675-1 |url=https://books.google.com/books?id=DJHI7DpgIbIC}}
• BOOK, Richard H., Crowell, Richard H. Crowell, Ralph, Fox, Ralph Fox, Introduction to Knot Theory, 1977, 978-0-387-90272-2,
• {{citation |first=Louis H. |last=Kauffman |authorlink=Louis H. Kauffman |title=On Knots |publisher= |location= |year=1987 |isbn=978-0-691-08435-0 |url=https://books.google.com/books?id=BLvGkIY8YzwC }}
• {{citation |first=Louis H. |last=Kauffman |authorlink=Louis H. Kauffman |title=Knots and Physics |publisher=World Scientific |year=2013 |isbn=978-981-4383-00-4 |edition=4th |url=https://books.google.com/books?id=fSKrRQ77FMkC }}

### Surveys

• {{citation |editor-first=William W. |editor-last=Menasco |editor2-first=Morwen |editor2-last=Thistlethwaite |editor2-link=Morwen Thistlethwaite |title=Handbook of Knot Theory |publisher=Elsevier |year=2005 |isbn=978-0-444-51452-3 }}
• Menasco and Thistlethwaite's handbook surveys a mix of topics relevant to current research trends in a manner accessible to advanced undergraduates but of interest to professional researchers.
• {{citation |last=Livio |first=Mario |chapter=Ch. 8: Unreasonable Effectiveness? |chapterurl=https://books.google.com/books?id=ebd6QofqY6QC&pg=PA203 |title=Is God a Mathematician? |publisher=Simon & Schuster |year=2009 |isbn=978-0-7432-9405-8 |pages=203â€“218 }}

{{commons category|Knot theory}}{{Wiktionary}}
• "Mathematics and Knots" This is an online version of an exhibition developed for the 1989 Royal Society "PopMath RoadShow". Its aim was to use knots to present methods of mathematics to the general public.

### History

• {{citation |last=Thomson |first=Sir William |authorlink=Lord Kelvin |title=On Vortex Atoms |journal=Proceedings of the Royal Society of Edinburgh |volume=VI |pages=94â€“105 |year=1867 |url=http://zapatopi.net/kelvin/papers/on_vortex_atoms.html}}
• {{citation |last=Silliman |first=Robert H. |title=William Thomson: Smoke Rings and Nineteenth-Century Atomism |journal=Isis |volume=54 |issue=4 |pages=461â€“474 |date=December 1963 |jstor=228151 |doi=10.1086/349764}}
• Movie of a modern recreation of Tait's smoke ring experiment
• History of knot theory (on the home page of Andrew Ranicki)

### Knot tables and software

{{Knot theory|state=collapsed}}

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