Knot
A knot is an intentional complication in cordage which may be practical or decorative, or both. Practical knots are classified by function, including hitches, bends, loop knots, and splices: a ''hitch'' fastens a rope to another object; a ' ...
s 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
In the mathematical field of topology, knot theory is the study of knot (mathematics), 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 ...
would arrive later with
Sir William Thomson
William Thomson, 1st Baron Kelvin, (26 June 182417 December 1907) was a British mathematician, mathematical physicist and engineer born in Belfast. Professor of Natural Philosophy at the University of Glasgow for 53 years, he did important ...
(Lord Kelvin) and his
vortex theory of the atom
The vortex theory of the atom was a 19th-century attempt by William Thomson (later Lord Kelvin) to explain why the atoms recently discovered by chemists came in only relatively few varieties but in very great numbers of each kind. Based on the ide ...
.
History
Pre-modern
Different knots are better at different tasks, such as
climbing
Climbing is the activity of using one's hands, feet, or any other part of the body to ascend a steep topographical object that can range from the world's tallest mountains (e.g. the eight thousanders), to small boulders. Climbing is done fo ...
or
sailing
Sailing employs the wind—acting on sails, wingsails or kites—to propel a craft on the surface of the ''water'' (sailing ship, sailboat, raft, windsurfer, or kitesurfer), on ''ice'' (iceboat) or on ''land'' (land yacht) over a chosen cour ...
. Knots were also regarded as having spiritual and religious symbolism in addition to their aesthetic qualities. The
endless knot
Endless knot in a Burmese Pali manuscript
The endless knot or eternal knot is a symbolic knot and one of the Eight Auspicious Symbols. It is an important symbol in Hinduism, Jainism and Buddhism. It is an important cultural marker in place ...
appears in Tibetan Buddhism, while the
Borromean rings
In mathematics, the Borromean rings are three simple closed curves in three-dimensional space that are topologically linked and cannot be separated from each other, but that break apart into two unknotted and unlinked loops when any one of the ...
have made repeated appearances in different cultures, often symbolizing unity. The
Celtic
Celtic, Celtics or Keltic may refer to:
Language and ethnicity
*pertaining to Celts, a collection of Indo-European peoples in Europe and Anatolia
**Celts (modern)
*Celtic languages
**Proto-Celtic language
* Celtic music
*Celtic nations
Sports Fo ...
monks who created the
Book of Kells
The Book of Kells ( la, Codex Cenannensis; ga, Leabhar Cheanannais; Dublin, Trinity College Library, MS A. I. 8 sometimes known as the Book of Columba) is an illuminated manuscript Gospel book in Latin, containing the four Gospels of the New ...
lavished entire pages with intricate
Celtic knot
Celtic knots ( ga, snaidhm Cheilteach, cy, cwlwm Celtaidd, kw, kolm Keltek, gd, snaidhm Ceilteach) are a variety of knots and stylized graphical representations of knots used for decoration, used extensively in the Celtic style of Insular ...
work.
Early modern
Knots were studied from a mathematical viewpoint by
Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
, who in 1833 developed the
Gauss linking integral
In mathematics, the linking number is a numerical invariant that describes the linking of two closed curves in three-dimensional space. Intuitively, the linking number represents the number of times that each curve winds around the other. In ...
for computing the
linking number
In mathematics, the linking number is a numerical invariant that describes the linking of two closed curves in three-dimensional space. Intuitively, the linking number represents the number of times that each curve winds around the other. In Eu ...
of two knots. His student
Johann Benedict Listing
Johann Benedict Listing (25 July 1808 – 24 December 1882) was a German mathematician.
J. B. Listing was born in Frankfurt and died in Göttingen. He first introduced the term "topology" to replace the older term "geometria situs" (also called ...
, after whom
Listing's knot is named, furthered their study.
In 1867 after observing
Scottish physicist
A physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe.
Physicists generally are interested in the root or ultimate caus ...
Peter Tait Peter Tait may refer to:
* Peter Tait (physicist) (1831–1901), Scottish mathematical physicist
* Peter Tait (footballer) (1936–1990), English professional footballer
* Peter Tait (mayor) (1915–1996), New Zealand politician
* Peter Tait (radio ...
's experiments involving smoke rings, Thomson came to the idea that atoms were knots of swirling vortices in the
æther. Chemical elements would thus correspond to knots and links. Tait's experiments were inspired by a paper of Helmholtz's on vortex-rings in incompressible fluids. Thomson and Tait believed that an understanding and classification of all possible knots would explain why atoms
absorb and emit light at only the discrete
wavelength
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tro ...
s that they do. For example, Thomson thought that sodium could be the
Hopf link
In mathematical knot theory, the Hopf link is the simplest nontrivial link with more than one component. It consists of two circles linked together exactly once, and is named after Heinz Hopf.
Geometric realization
A concrete model consists of ...
due to its two lines of spectra.
[Alexei Sossinsky (2002) ''Knots, Mathematics with a Twist'', ]Harvard University Press
Harvard University Press (HUP) is a publishing house established on January 13, 1913, as a division of Harvard University, and focused on academic publishing. It is a member of the Association of American University Presses. After the retirem ...
Tait subsequently began listing unique knots in the belief that he was creating a table of elements. He formulated what is now known as the
Tait conjectures
The Tait conjectures are three conjectures made by 19th-century mathematician Peter Guthrie Tait in his study of knots.. The Tait conjectures involve concepts in knot theory such as alternating knots, chirality, and writhe. All of the Tait conje ...
on
alternating knot
In knot theory, a knot or link diagram is alternating if the crossings alternate under, over, under, over, as one travels along each component of the link. A link is alternating if it has an alternating diagram.
Many of the knots with crossing ...
s. (The conjectures were proved in the 1990s.) Tait's knot tables were subsequently improved upon by
C. N. Little and
Thomas Kirkman
Thomas Penyngton Kirkman FRS (31 March 1806 – 3 February 1895) was a British mathematician and ordained minister of the Church of England. Despite being primarily a churchman, he maintained an active interest in research-level mathematics, a ...
.
[
]James Clerk Maxwell
James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and ligh ...
, a colleague and friend of Thomson's and Tait's, also developed a strong interest in knots. Maxwell studied Listing's work on knots. He re-interpreted Gauss's linking integral in terms of electromagnetic theory. In his formulation, the integral represented the work done by a charged particle moving along one component of the link under the influence of the magnetic field generated by an electric current along with the other component. Maxwell also continued the study of smoke rings by considering three interacting rings.
When the ''luminiferous æther'' was not detected in the Michelson–Morley experiment, vortex theory became completely obsolete, and knot theory ceased to be of great scientific interest. Modern physics demonstrates that the discrete wavelengths depend on quantum energy level
A quantum mechanical system or particle that is bound—that is, confined spatially—can only take on certain discrete values of energy, called energy levels. This contrasts with classical particles, which can have any amount of energy. The te ...
s.
Late Modern
Following the development topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
in the early 20th century spearheaded by Henri Poincaré
Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The ...
, topologists such as Max Dehn
Max Wilhelm Dehn (November 13, 1878 – June 27, 1952) was a German mathematician most famous for his work in geometry, topology and geometric group theory. Born to a Jewish family in Germany, Dehn's early life and career took place in Germany. ...
, J. W. Alexander, and Kurt Reidemeister
Kurt Werner Friedrich Reidemeister (13 October 1893 – 8 July 1971) was a mathematician born in Braunschweig (Brunswick), Germany.
Life
He was a brother of Marie Neurath.
Beginning in 1912, he studied in Freiburg, Munich, Marburg, and Göttinge ...
, investigated knots. Out of this sprang the Reidemeister move
Kurt Werner Friedrich Reidemeister (13 October 1893 – 8 July 1971) was a mathematician born in Braunschweig (Brunswick), Germany.
Life
He was a brother of Marie Neurath.
Beginning in 1912, he studied in Freiburg, Munich, Marburg, and Götting ...
s and the Alexander polynomial
In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander II discovered this, the first knot polynomial, in 1923. In 1969, John Conway showed a ve ...
.[ Dehn also developed ]Dehn surgery
In topology, a branch of mathematics, a Dehn surgery, named after Max Dehn, is a construction used to modify 3-manifolds. The process takes as input a 3-manifold together with a link. It is often conceptualized as two steps: ''drilling'' then '' ...
, which related knots to the general theory of 3-manifolds, and formulated the Dehn problems in group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
, such as the word problem. Early pioneers in the first half of the 20th century include Ralph Fox
Ralph Hartzler Fox (March 24, 1913 – December 23, 1973) was an American mathematician. As a professor at Princeton University, he taught and advised many of the contributors to the ''Golden Age of differential topology'', and he played ...
, who popularized the subject. In this early period, knot theory primarily consisted of study into the knot group
In mathematics, a knot is an embedding of a circle into 3-dimensional Euclidean space. The knot group of a knot ''K'' is defined as the fundamental group of the knot complement of ''K'' in R3,
:\pi_1(\mathbb^3 \setminus K).
Other conventions co ...
and homological
Homology may refer to:
Sciences
Biology
*Homology (biology), any characteristic of biological organisms that is derived from a common ancestor
*Sequence homology, biological homology between DNA, RNA, or protein sequences
*Homologous chromo ...
invariants of the knot complement
In mathematics, the knot complement of a tame knot ''K'' is the space where the knot is not. If a knot is embedded in the 3-sphere, then the complement is the 3-sphere minus the space near the knot. To make this precise, suppose that ''K'' is a ...
.
Contemporary
In 1961 Wolfgang Haken
Wolfgang Haken (June 21, 1928 – October 2, 2022) was a German American mathematician who specialized in topology, in particular 3-manifolds.
Biography
Haken was born in Berlin, Germany. His father was Werner Haken, a physicist who had Max ...
discovered an algorithm that can determine whether or not a knot is non-trivial. He also outlined a strategy for solving the general knot recognition problem, i.e. determining if two given knots are equivalent or not. In the early 1970s, Friedhelm Waldhausen
Friedhelm Waldhausen (born 1938 in Millich, Hückelhoven, Rhine Province) is a German mathematician known for his work in algebraic topology. He made fundamental contributions in the fields of 3-manifolds and (algebraic) K-theory.
Career
Wald ...
announced the completion of Haken's program based on his results and those of Klaus Johannson
Klaus is a German, Dutch and Scandinavian given name and surname. It originated as a short form of Nikolaus, a German form of the Greek given name Nicholas.
Notable persons whose family name is Klaus
* Billy Klaus (1928–2006), American baseb ...
, William Jaco
William "Bus" H. Jaco (born July 14, 1940 in Grafton, West Virginia) is an American mathematician who is known for his role in the Jaco–Shalen–Johannson decomposition theorem and is currently Regents Professor and Grayce B. Kerr Chair at Okl ...
, Peter Shalen
Peter B. Shalen (born c. 1946) is an American mathematician, working primarily in low-dimensional topology. He is the "S" in JSJ decomposition.
Life
He graduated from Stuyvesant High School in 1962, and went on to earn a B.A. from Harvard Colle ...
, and Geoffrey Hemion. In 2003 Sergei Matveev pointed out and filled in a crucial gap.
A few major discoveries in the late 20th century greatly rejuvenated knot theory and brought it further into the mainstream. In the late 1970s William Thurston
William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds.
Thurston ...
's hyperbolization theorem
In geometry, Thurston's geometrization theorem or hyperbolization theorem implies that closed atoroidal Haken manifolds are hyperbolic, and in particular satisfy the Thurston conjecture.
Statement
One form of Thurston's geometrization theore ...
introduced the theory of hyperbolic 3-manifold
In mathematics, more precisely in topology and differential geometry, a hyperbolic 3–manifold is a manifold of dimension 3 equipped with a hyperbolic metric, that is a Riemannian metric which has all its sectional curvatures equal to -1. It ...
s into knot theory and made it of prime importance. In 1982, Thurston received a Fields Medal, the highest honor in mathematics, largely due to this breakthrough. Thurston's work also led, after much expansion by others, to the effective use of tools from representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
and algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
. Important results followed, including the Gordon–Luecke theorem
In mathematics, the Gordon–Luecke theorem on knot complements states that if the complements of two tame knots are homeomorphic, then the knots are equivalent. In particular, any homeomorphism between knot complements must take a meridian to a me ...
, which showed that knots were determined (up to mirror-reflection) by their complements, and the Smith conjecture
In mathematics, the Smith conjecture states that if ''f'' is a diffeomorphism of the 3-sphere of Order (group theory), finite order, then the fixed point set of ''f'' cannot be a nontrivial knot (mathematics), knot.
showed that a non-trivial or ...
.
Interest in knot theory from the general mathematical community grew significantly after Vaughan Jones
Sir Vaughan Frederick Randal Jones (31 December 19526 September 2020) was a New Zealand mathematician known for his work on von Neumann algebras and knot polynomials. He was awarded a Fields Medal in 1990.
Early life
Jones was born in Gisb ...
' discovery of the Jones polynomial
In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984. Specifically, it is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polynom ...
in 1984. This led to other knot polynomials such as the bracket polynomial In the mathematical field of knot theory, the bracket polynomial (also known as the Kauffman bracket) is a polynomial invariant of framed links. Although it is not an invariant of knots or links (as it is not invariant under type I Reidemeister mov ...
, HOMFLY polynomial
In the mathematical field of knot theory, the HOMFLY polynomial or HOMFLYPT polynomial, sometimes called the generalized Jones polynomial, is a 2-variable knot polynomial, i.e. a knot invariant in the form of a polynomial of variables ''m'' and ' ...
, and Kauffman polynomial
In knot theory, the Kauffman polynomial is a 2-variable knot polynomial due to Louis Kauffman. It is initially defined on a link diagram as
:F(K)(a,z)=a^L(K)\,,
where w(K) is the writhe of the link diagram and L(K) is a polynomial in ''a'' and ' ...
. Jones was awarded the Fields medal
The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of the International Mathematical Union (IMU), a meeting that takes place every four years. The name of the award ho ...
in 1990 for this work.[ In 1988 ]Edward Witten
Edward Witten (born August 26, 1951) is an American mathematical and theoretical physicist. He is a Professor Emeritus in the School of Natural Sciences at the Institute for Advanced Study in Princeton. Witten is a researcher in string theory, q ...
proposed a new framework for the Jones polynomial, utilizing existing ideas from mathematical physics
Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and t ...
, such as Feynman path integral
The path integral formulation is a description in quantum mechanics that generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional i ...
s, and introducing new notions such as topological quantum field theory
In gauge theory and mathematical physics, a topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which computes topological invariants.
Although TQFTs were invented by physicists, they are also of mathem ...
. Witten also received the Fields medal, in 1990, partly for this work. Witten's description of the Jones polynomial implied related invariants for 3-manifold
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds lo ...
s. Simultaneous, but different, approaches by other mathematicians resulted in the Witten–Reshetikhin–Turaev invariants and various so-called "quantum invariant
In the mathematical field of knot theory, a quantum knot invariant or quantum invariant of a knot or link is a linear sum of colored Jones polynomial of surgery presentations of the knot complement.
List of invariants
*Finite type invariant
*Kont ...
s", which appear to be the mathematically rigorous version of Witten's invariants . In the 1980s John Horton Conway
John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches ...
discovered a procedure for unknotting knots gradually known as Conway notation.
In 1992, the ''Journal of Knot Theory and Its Ramifications
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 creat ...
'' was founded, establishing a journal devoted purely to knot theory.
In the early 1990s, knot invariants which encompass the Jones polynomial and its generalizations, called the finite type invariant In the mathematical theory of knots, a finite type invariant, or Vassiliev invariant (so named after Victor Anatolyevich Vassiliev), is a knot invariant that can be extended (in a precise manner to be described) to an invariant of certain singular ...
s, were discovered by Vassiliev and Goussarov. These invariants, initially described using "classical" topological means, were shown by 1994 Fields Medalist Maxim Kontsevich
Maxim Lvovich Kontsevich (russian: Макси́м Льво́вич Конце́вич, ; born 25 August 1964) is a Russian and French mathematician and mathematical physicist. He is a professor at the Institut des Hautes Études Scientifiques an ...
to result from integration
Integration may refer to:
Biology
*Multisensory integration
*Path integration
* Pre-integration complex, viral genetic material used to insert a viral genome into a host genome
*DNA integration, by means of site-specific recombinase technology, ...
, using the Kontsevich integral In the mathematical theory of knots, the Kontsevich invariant, also known as the Kontsevich integral of an oriented framed link, is a universal Vassiliev invariant in the sense that any coefficient of the Kontsevich invariant is of a finite type ...
, of certain algebraic structures (, ).
These breakthroughs were followed by the discovery of Khovanov homology In mathematics, Khovanov homology is an oriented link invariant that arises as the cohomology of a cochain complex. It may be regarded as a categorification of the Jones polynomial.
It was developed in the late 1990s by Mikhail Khovanov, then at ...
and knot Floer homology, which greatly generalize the Jones and Alexander polynomials. These homology theories have contributed to further mainstreaming of knot theory.
In the last several decades of the 20th century, scientists and mathematicians began finding applications of knot theory to problems in biology
Biology is the scientific study of life. It is a natural science with a broad scope but has several unifying themes that tie it together as a single, coherent field. For instance, all organisms are made up of cells that process hereditary i ...
and chemistry
Chemistry is the science, scientific study of the properties and behavior of matter. It is a natural science that covers the Chemical element, elements that make up matter to the chemical compound, compounds made of atoms, molecules and ions ...
. Knot theory can be used to determine if a molecule is chiral
Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object.
An object or a system is ''chiral'' if it is distinguishable from ...
(has a "handedness") or not. Chemical compounds of different handedness can have drastically differing properties, thalidomide
Thalidomide, sold under the brand names Contergan and Thalomid among others, is a medication used to treat a number of cancers (including multiple myeloma), graft-versus-host disease, and a number of skin conditions including complications of ...
being a notable example of this. More generally, knot theoretic methods have been used in studying topoisomer Topoisomers or topological isomers are molecules with the same chemical formula and stereochemical bond connectivities but different topologies. Examples of molecules for which there exist topoisomers include DNA, which can form knots, and catenan ...
s, topologically different arrangements of the same chemical formula. The closely related theory of tangle
Tangle may refer to: Science, Technology, Engineering & Mathematics
*''The Tangle'' is the name of the ledger, a directed acyclic graph, used for the cryptocurrency IOTA
*Tangle (mathematics), a topological object
Natural sciences & medicine ...
s have been effectively used in studying the action of certain enzymes on DNA. The interdisciplinary field of physical knot theory
Physical may refer to:
*Physical examination
In a physical examination, medical examination, or clinical examination, a medical practitioner examines a patient for any possible medical signs or symptoms of a medical condition. It generally cons ...
investigates mathematical models of knots based on physical considerations in order to understand knotting phenomena arising in materials like DNA or polymers.
In physics it has been shown that certain hypothetical quasiparticle
In physics, quasiparticles and collective excitations are closely related emergent phenomena arising when a microscopically complicated system such as a solid behaves as if it contained different weakly interacting particles in vacuum.
For exam ...
s such as nonabelian anyon
In physics, an anyon is a type of quasiparticle that occurs only in two-dimensional systems, with properties much less restricted than the two kinds of standard elementary particles, fermions and bosons. In general, the operation of exchangi ...
s exhibit useful topological properties, namely that their quantum state
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in ...
s are left unchanged by ambient isotopy
In the mathematical subject of topology, an ambient isotopy, also called an ''h-isotopy'', is a kind of continuous distortion of an ambient space, for example a manifold, taking a submanifold to another submanifold. For example in knot theory, one ...
of their world line
The world line (or worldline) of an object is the path that an object traces in 4-dimensional spacetime. It is an important concept in modern physics, and particularly theoretical physics.
The concept of a "world line" is distinguished from con ...
s. It is hoped that they can be used to make a quantum computer
Quantum computing is a type of computation whose operations can harness the phenomena of quantum mechanics, such as superposition, interference, and entanglement. Devices that perform quantum computations are known as quantum computers. Though ...
resistant to decohere
Quantum decoherence is the loss of quantum coherence. In quantum mechanics, particles such as electrons are described by a wave function, a mathematical representation of the quantum state of a system; a probabilistic interpretation of the wave ...
nce. Since the world lines form a mathematical ''braid'', braid theory
A braid (also referred to as a plait) is a complex structure or pattern formed by interlacing two or more strands of flexible material such as textile yarns, wire, or hair.
The simplest and most common version is a flat, solid, three-strande ...
, a related field to knot theory
In the mathematical field of topology, knot theory is the study of knot (mathematics), 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 ...
, is used in studying the properties of such a computer, called a topological quantum computer
A topological quantum computer is a theoretical quantum computer proposed by Russian-American physicist Alexei Kitaev in 1997. It employs quasiparticles in two-dimensional systems, called anyons, whose world lines pass around one another to for ...
.
See also
* Quantum knots
Notes
{{Reflist
References
* Silver, Dan,
Scottish physics and knot theory's odd origins
' (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
World Scientific Publishing is an academic publisher of scientific, technical, and medical books and journals headquartered in Singapore. The company was founded in 1981. It publishes about 600 books annually, along with 135 journals in various f ...
.
External links
* 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
of a modern recreation of Tait's smoke ring experiment
Knot theory