Knot theory

trivial knot (top left) and the trefoil knot

(below it)

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 so 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, ${\displaystyle \mathbb {E} ^{3}}$. Two mathematical knots are equivalent if one can be transformed into the other via a deformation of ${\displaystyle \mathbb {R} ^{3}}$ upon itself (known as an ambient isotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting it or passing it through itself.

Knots can be described in various ways. Using different description methods, there may be more than one description of the same knot. For example, a common method of describing a knot is a planar diagram called a knot diagram, in which any knot can be drawn in many different ways. 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

, 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). For example, a higher-dimensional knot is an n-dimensional sphere embedded in (n+2)-dimensional Euclidean space.

Archaeologists have discovered that knot tying dates back to prehistoric times. Besides their uses such as

.

A mathematical theory of knots was first developed in 1771 by

.

These topologists in the early part of the 20th century—Max Dehn, J. W. Alexander, and others—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,

.

In the last several decades of the 20th century, scientists became interested in studying

).

Knot equivalence

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 (Adams 2004) (Sossinsky 2002). Simply, we can say a knot ${\displaystyle K}$ is a "simple closed curve" (see

${\displaystyle K\colon [0,1]\to \mathbb {R} ^{3}}$, with the only "non-injectivity" being ${\displaystyle K(0)=K(1)}$. Topologists consider knots and other entanglements such as

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 ${\displaystyle K_{1},K_{2}}$ are equivalent if there is an

${\displaystyle h\colon \mathbb {R} ^{3}\to \mathbb {R} ^{3}}$ with ${\displaystyle h(K_{1})=K_{2}}$.

What this definition of knot equivalence means is that two knots are equivalent when there is a continuous family of homeomorphisms ${\displaystyle \{h_{t}:\mathbb {R} ^{3}\rightarrow \mathbb {R} ^{3}\ \mathrm {for} \ 0\leq t\leq 1\}}$ of space onto itself, such that the last one of them carries the first knot onto the second knot. (In detail: Two knots ${\displaystyle K_{1}}$ and ${\displaystyle K_{2}}$ are equivalent if there exists a continuous mapping ${\displaystyle H:\mathbb {R} ^{3}\times [0,1]\rightarrow \mathbb {R} ^{3}}$ such that a) for each ${\displaystyle t\in [0,1]}$ the mapping taking ${\displaystyle x\in \mathbb {R} ^{3}}$ to ${\displaystyle H(x,t)\in \mathbb {R} ^{3}}$ is a homeomorphism of ${\displaystyle \mathbb {R} ^{3}}$ onto itself; b) ${\displaystyle H(x,0)=x}$ for all ${\displaystyle x\in \mathbb {R} ^{3}}$; and c) ${\displaystyle H(K_{1},1)=K_{2}}$. Such a function ${\displaystyle H}$ 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 ${\displaystyle \mathbb {R} ^{3}}$ 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 ${\displaystyle t=1}$ (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 (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 (Hass 1998). The special case of recognizing the unknot, called the unknotting problem, is of particular interest (Hoste 2005). In February 2021 Marc Lackenby announced a new unknot recognition algorithm that runs in quasi-polynomial time.^[2]

A useful way to visualise and manipulate knots is to project the knot onto a plane—think of the knot casting a shadow on the wall. A small change in the direction of projection will ensure that it is

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.^[3][4] A petal projection is a type of projection in which, instead of forming double points, all strands of the knot meet at a single crossing point, connected to it by loops forming non-nested "petals".^[5]

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:

Reidemeister moves

Type I	Type II
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 (Sossinsky 2002, ch. 3) (Lickorish 1997, ch. 1).

A knot invariant is a "quantity" that is the same for equivalent knots (Adams 2004) (Lickorish 1997) (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

and hyperbolic invariants were discovered. These aforementioned invariants are only the tip of the iceberg of modern knot theory.

A knot polynomial is a

).

The Alexander–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 ${\displaystyle L_{+},L_{-},L_{0}}$ be the oriented link diagrams resulting from changing the diagram as indicated in the figure:

The original diagram might be either ${\displaystyle L_{+}}$ or ${\displaystyle L_{-}}$, depending on the chosen crossing's configuration. Then the Alexander–Conway polynomial, ${\displaystyle C(z)}$, is recursively defined according to the rules:

• ${\displaystyle C(O)=1}$ (where ${\displaystyle O}$ is any diagram of the unknot)
• ${\displaystyle C(L_{+})=C(L_{-})+zC(L_{0}).}$

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 Alexander–Conway polynomial of the trefoil knot. The yellow patches indicate where the relation is applied.

C() = C() + z C()

gives the unknot and the Hopf link. Applying the relation to the Hopf link where indicated,

C() = C() + z C()

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:

C() = C() + z C()

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:

${\displaystyle C(\mathrm {trefoil} )=1+z(0+z)=1+z^{2}}$

Since the Alexander–Conway polynomial is a knot invariant, this shows that the trefoil is not equivalent to the unknot. So the trefoil really is "knotted".

• 
  The left-handed trefoil knot.
• 
  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 amphichiral. This was shown by Max Dehn, before the invention of knot polynomials, using group theoretical methods (Dehn 1914). But the Alexander–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 (Lickorish 1997).

).

Geometry lets us visualize what the inside of a knot or link complement looks like by imagining light rays as traveling along the

neighborhoods of the link. By thickening the link in a standard way, the horoball neighborhoods of the link components are obtained. Even though the boundary of a neighborhood is a torus, when viewed from inside the link complement, it looks like a sphere. Each link component shows up as infinitely many spheres (of one color) as there are infinitely many light rays from the observer to the link component. The fundamental parallelogram (which is indicated in the picture), tiles both vertically and horizontally and shows how to extend the pattern of spheres infinitely.

This pattern, the horoball pattern, is itself a useful invariant. Other hyperbolic invariants include the shape of the fundamental parallelogram, length of shortest geodesic, and volume. Modern knot and link tabulation efforts have utilized these invariants effectively. Fast computers and clever methods of obtaining these invariants make calculating these invariants, in practice, a simple task (Adams, Hildebrand & Weeks 1991).

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.

Four-dimensional space occurs in classical knot theory, however, and an important topic is the study of slice knots and ribbon knots. A notorious open problem asks whether every slice knot is also ribbon.

Since a knot can be considered topologically a 1-dimensional sphere, the next generalization is to consider a

(${\displaystyle \mathbb {S} ^{2}}$) embedded in 4-dimensional Euclidean space (${\displaystyle \mathbb {R} ^{4}}$). Such an embedding is knotted if there is no homeomorphism of ${\displaystyle \mathbb {R} ^{4}}$ 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 (Zeeman 1963), although this is no longer a requirement for smoothly knotted spheres. In fact, there are smoothly knotted ${\displaystyle (4k-1)}$-spheres in 6k-dimensional space; e.g., there is a smoothly knotted 3-sphere in ${\displaystyle \mathbb {R} ^{6}}$ (Haefliger 1962) (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 ${\displaystyle \mathbb {R} ^{n}}$ with ${\displaystyle 2n-3k-3>0}$ is unknotted. The notion of a knot has further generalisations in mathematics, see: Knot (mathematics), isotopy classification of embeddings.

Every knot in the n-sphere ${\displaystyle \mathbb {S} ^{n}}$ is the link of a

with isolated singularity in ${\displaystyle \mathbb {R} ^{n+1}}$ (Akbulut & King 1981).

An n-knot is a single ${\displaystyle \mathbb {S} ^{n}}$ embedded in ${\displaystyle \mathbb {R} ^{m}}$. An n-link consists of k-copies of ${\displaystyle \mathbb {S} ^{n}}$ embedded in ${\displaystyle \mathbb {R} ^{m}}$, where k is a natural number. Both the ${\displaystyle m=n+2}$ and the ${\displaystyle m>n+2}$ cases are well studied, and so is the ${\displaystyle n>1}$ case.^[6][7]

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 (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

). 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.

Knots can also be constructed using the circuit topology approach. This is done by combining basic units called soft contacts using five operations (Parallel, Series, Cross, Concerted, and Sub).^[8][9] The approach is applicable to open chains as well and can also be extended to include the so-called hard contacts.

Traditionally, knots have been catalogued in terms of crossing number. Knot tables generally include only prime knots, and only one entry for a knot and its mirror image (even if they are different) (Hoste, Thistlethwaite & Weeks 1998). The number of nontrivial knots of a given crossing number increases rapidly, making tabulation computationally difficult (Hoste 2005, p. 20). Tabulation efforts have succeeded in enumerating over 6 billion knots and links (Hoste 2005, p. 28). The sequence of the number of prime knots of a given crossing number, up to crossing number 16, is 0, 0, 1, 1, 2, 3, 7, 21, 49, 165, 552, 2176, 9988, 46972, 253293, 1388705... (sequence A002863 in the OEIS). While exponential upper and lower bounds for this sequence are known, it has not been proven that this sequence is strictly increasing (Adams 2004).

The first knot tables by Tait, Little, and Kirkman used knot diagrams, although Tait also used a precursor to the

).

The early tables attempted to list all knots of at most 10 crossings, and all alternating knots of 11 crossings (Hoste, Thistlethwaite & Weeks 1998). The development of knot theory due to Alexander, Reidemeister, Seifert, and others eased the task of verification and tables of knots up to and including 9 crossings were published by Alexander–Briggs and Reidemeister in the late 1920s.

The first major verification of this work was done in the 1960s by

. [see Perko (1982), Primality of certain knots, Topology Proceedings] Less famous is the duplicate in his 10 crossing link table: 2.-2.-20.20 is the mirror of 8*-20:-20. [See Perko (2016), Historical highlights of non-cyclic knot theory, J. Knot Theory Ramifications].

In the late 1990s Hoste, Thistlethwaite, and Weeks tabulated all the knots through 16 crossings (Hoste, Thistlethwaite & Weeks 1998). In 2003 Rankin, Flint, and Schermann, tabulated the alternating knots through 22 crossings (Hoste 2005). In 2020 Burton tabulated all prime knots with up to 19 crossings (Burton 2020).

This is the most traditional notation, due to the 1927 paper of

Dowker–Thistlethwaite notation

The Dowker–Thistlethwaite notation, also called the Dowker 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–Thistlethwaite 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–Thistlethwaite notation.

The Conway notation for knots and links, named after John Horton Conway, is based on the theory of tangles (Conway 1970). The advantage of this notation is that it reflects some properties of the knot or link.

The notation describes how to construct a particular link diagram of the link. Start with a basic polyhedron, a 4-valent connected planar graph with no digon regions. Such a polyhedron is denoted first by the number of vertices then a number of asterisks which determine the polyhedron's position on a list of basic polyhedra. For example, 10** denotes the second 10-vertex polyhedron on Conway's list.

Each vertex then has an

substituted into it (each vertex is oriented so there is no arbitrary choice in substitution). Each such tangle has a notation consisting of numbers and + or − signs.

An example is 1*2 −3 2. The 1* denotes the only 1-vertex basic polyhedron. The 2 −3 2 is a sequence describing the continued fraction associated to a

. One inserts this tangle at the vertex of the basic polyhedron 1*.

A more complicated example is 8*3.1.2 0.1.1.1.1.1 Here again 8* refers to a basic polyhedron with 8 vertices. The periods separate the notation for each tangle.

Any link admits such a description, and it is clear this is a very compact notation even for very large crossing number. There are some further shorthands usually used. The last example is usually written 8*3:2 0, where the ones are omitted and kept the number of dots excepting the dots at the end. For an algebraic knot such as in the first example, 1* is often omitted.

Conway's pioneering paper on the subject lists up to 10-vertex basic polyhedra of which he uses to tabulate links, which have become standard for those links. For a further listing of higher vertex polyhedra, there are nonstandard choices available.

, similar to the Dowker–Thistlethwaite 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,−2,3,−1,2,−3

Gauss code is limited in its ability to identify knots. This problem is partially addressed with by the

.

• Arithmetic rope
• Circuit topology
• Lamp cord trick
• Legendrian submanifolds and knots
• List of knot theory topics
• Molecular knot
• Necktie § Knots
• Quantum topology
• Ribbon theory

• ISBN 978-0-8218-3678-1
• Adams, Colin; Crawford, Thomas; DeMeo, Benjamin; Landry, Michael; Lin, Alex Tong; Montee, MurphyKate; Park, Seojung; Venkatesh, Saraswathi; Yhee, Farrah (2015), "Knot projections with a single multi-crossing", Journal of Knot Theory and Its Ramifications, 24 (3): 1550011, 30,
  S2CID 119320887
• Adams, Colin; Hildebrand, Martin;
  JSTOR 2001854
• S2CID 120218312
• doi:10.1016/0040-9383(95)93237-2
• Burton, Benjamin A. (2020). "The Next 350 Million Knots". 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz Int. Proc. Inform. Vol. 164. Schloss Dagstuhl–Leibniz-Zentrum für Informatik. pp. 25:1–25:17.
  doi:10.4230/LIPIcs.SoCG.2020.25
  .
• Collins, Graham (April 2006), "Computing with Quantum Knots",
  PMID 16596880
• S2CID 120452571
• ISBN 978-0-08-012975-4
• Doll, Helmut; Hoste, Jim (1991), "A tabulation of oriented links. With microfiche supplement", Math. Comp., 57 (196): 747–761,
  doi:10.1090/S0025-5718-1991-1094946-4
• ISBN 978-0-521-66254-3
• JSTOR 1970208
• MR 0160196
• S2CID 7381505
• Hoste, Jim;
  S2CID 18027155
• Hoste, Jim (2005). "The Enumeration and Classification of Knots and Links". Handbook of Knot Theory. pp. 209–232.
  ISBN 978-0-444-51452-3
  .
• JSTOR 1970561
• Kontsevich, M. (1993). "Vassiliev's knot invariants". I. M. Gelfand Seminar. ADVSOV. Vol. 16. pp. 137–150.
  ISBN 978-0-8218-4117-4
  .
• S2CID 122824389
• Perko, Kenneth (1974), "On the classification of knots",
  JSTOR 2040074
• Rolfsen, Dale (1976), Knots and Links, Mathematics Lecture Series, vol. 7,
  MR 0515288
• Schubert, Horst (1949). Die eindeutige Zerlegbarkeit eines Knotens in Primknoten.
  ISBN 978-3-540-01419-5
  .
• Silver, Daniel (2006). "Knot Theory's Odd Origins". American Scientist. 94 (2): 158.
  doi:10.1511/2006.2.158
  .
• Simon, Jonathan (1986), "Topological chirality of certain molecules", Topology, 25 (2): 229–235,
  doi:10.1016/0040-9383(86)90041-8
• Sossinsky, Alexei (2002), Knots, mathematics with a twist, Harvard University Press,
  ISBN 978-0-674-00944-8
• Turaev, Vladimir G. (2016). Quantum Invariants of Knots and 3-Manifolds.
  S2CID 118682559
  .
• Weisstein, Eric W. (2013). "Reduced Knot Diagram". MathWorld. Wolfram. Retrieved 8 May 2013.
• Weisstein, Eric W. (2013a). "Reducible Crossing". MathWorld. Wolfram. Retrieved 8 May 2013.
• S2CID 14951363
• JSTOR 1970538

Footnotes

1. ^ As first sketched using the theory of Haken manifolds by Haken (1962). For a more recent survey, see Hass (1998)
2. ^ Marc Lackenby announces a new unknot recognition algorithm that runs in quasi-polynomial time, Mathematical Institute, University of Oxford, 2021-02-03, retrieved 2021-02-03
3. ^ Weisstein 2013.
4. ^ Weisstein 2013a.
5. ^ Adams et al. 2015.
6. ISBN 978-0691049380
   — An introductory article to high dimensional knots and links for the advanced readers
7. Bibcode:2013arXiv1304.6053O
   — An introductory article to high dimensional knots and links for beginners
8. doi:10.3390/sym13122353
   .
9. PMID 37264056
   .
10. ^ "The Revenge of the Perko Pair", RichardElwes.co.uk. Accessed February 2016. Richard Elwes points out a common mistake in describing the Perko pair.

Further reading

Introductory textbooks

There are a number of introductions to knot theory. A classical introduction for graduate students or advanced undergraduates is (Rolfsen 1976). Other good texts from the references are (Adams 2004) and (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. (Cromwell 2004) is suitable for undergraduates who know point-set topology; knowledge of algebraic topology is not required.

• ISBN 978-3-11-008675-1
• ISBN 978-0-387-90272-2
  .
• ISBN 978-0-691-08435-0
• ISBN 978-981-4383-00-4
• Cromwell, Peter R. (2004), Knots and Links, Cambridge University Press,
  ISBN 978-0-521-54831-1

Surveys

• Menasco, William W.;
  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.
• Livio, Mario (2009), "Ch. 8: Unreasonable Effectiveness?", Is God a Mathematician?, Simon & Schuster, pp. 203–218,
  ISBN 978-0-7432-9405-8

External links

Wikimedia Commons has media related to Knot theory.

Look up knot theory in Wiktionary, the free dictionary.

• "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

• Thomson, Sir William (1867), "On Vortex Atoms", Proceedings of the Royal Society of Edinburgh, VI: 94–105
• Silliman, Robert H. (December 1963), "William Thomson: Smoke Rings and Nineteenth-Century Atomism", Isis, 54 (4): 461–474,
  S2CID 144988108
• 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

• KnotInfo: Table of Knot Invariants and Knot Theory Resources
• The Knot Atlas — detailed info on individual knots in knot tables
• KnotPlot — software to investigate geometric properties of knots
• Knotscape — software to create images of knots
• Knoutilus — online database and image generator of knots
• KnotData.html — Wolfram Mathematica function for investigating knots
• Regina — software for low-dimensional topology with native support for knots and links. Tables of prime knots with up to 19 crossings