History of knot theory
Encyclopedia
For thousands of years, knots have been used for basic purposes such as recording information, fastening and tying objects together. Over time people realized that different knots were better at different tasks, such as climbing
or sailing
. Knots were also regarded as having spiritual and religious symbolism in addition to their aesthetic qualities. The endless knot
appears in Tibetan Buddhism, while the Borromean rings
have made repeated appearances in different cultures, often symbolizing unity. The Celtic
monks who created the Book of Kells
lavished entire pages with intricate Celtic knot
work.
Knots were studied from a mathematical viewpoint by Carl Friedrich Gauss
, who in 1833 developed the Gauss linking integral for computing the linking number
of two knots. His student Johann Benedict Listing
, after whom Listing's knot
is named, furthered their study.
The early, significant stimulus in knot theory
would arrive later with Sir William Thomson (Lord Kelvin) and his theory of vortex atoms.
In 1867 after observing Scottish
physicist
Peter Tait
'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
s that they do. For example, Thomson thought that sodium could be the Hopf link
due to its two lines of spectra.
Tait subsequently began listing unique knots in the belief that he was creating a table of elements. He formulated what are now known as the Tait conjectures
on alternating knot
s. (The conjectures were finally resolved in the 1990s.) Tait's knot tables were subsequently improved upon by C. N. Little and T. P. Kirkman.
James Clerk Maxwell
, 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' 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 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 levels.
Following the development of topology
in the early 20th century spearheaded by Henri Poincaré
, topologists such as Max Dehn
, J. W. Alexander
, and Kurt Reidemeister
, investigated knots. Out of this sprang the Reidemeister move
s and the Alexander polynomial
. Dehn also developed Dehn surgery
, which related knots to the general theory of 3-manifolds, and formulated the Dehn problems in group theory
, such as the word problem
. Early pioneers in the first half of the 20th century include Ralph Fox
, who popularized the subject. In this early period, knot theory primarily consisted of study into the knot group
and homological
invariants of the knot complement
.
In 1961 Wolfgang Haken
discovered an algorithm that can determine whether or not a knot is a knot
. 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
announced the completion of Haken's program based on his results and those of Klaus Johannson, William Jaco
, Peter Shalen
, 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
's hyperbolization theorem
introduced the theory of hyperbolic 3-manifold
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
and algebraic geometry
. Important results followed, including the Gordon-Luecke theorem
, which showed that knots were determined (up to mirror-reflection) by their complements, and the Smith conjecture
.
Interest in knot theory from the general mathematical community grew significantly after Vaughan Jones
' discovery of the Jones polynomial in 1984. This led to other knot polynomials such as the bracket polynomial
, HOMFLY polynomial
, and Kauffman polynomial. Jones was awarded the Fields medal
in 1990 for this work. In 1988 Edward Witten
proposed a new framework for the Jones polynomial, utilizing existing ideas from mathematical physics
, such as Feynman path integrals, and introducing new notions such as topological quantum field theory
. 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
s. Simultaneous, but different, approaches by other mathematicians resulted in the Witten-Reshetikhin-Turaev invariants and various so-called "quantum invariant
s", which appear to be the mathematically rigorous version of Witten's invariants .
In 1992, the Journal of Knot Theory and Its Ramifications
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
s, were discovered by Vassiliev and Goussarov
. These invariants, initially described using "classical" topological means, were shown by 1994 Fields Medalist Maxim Kontsevich
to result from integration
, using the Kontsevich integral, of certain algebraic structures .
These breakthroughs were followed by the discovery of Khovanov homology
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
and chemistry
. Knot theory can be used to determine if a molecule is chiral
(has a "handedness") or not. Chemical compounds of different handedness can have drastically differing properties, thalidomide being a notable example of this. More generally, knot theoretic methods have been used in studying topoisomer
s, topologically different arrangements of the same chemical formula. The closely related theory of tangles have been effectively used in studying the action of certain enzymes on DNA. The interdisciplinary field of physical knot theory
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
s such as nonabelian anyon
s exhibit useful topological properties, namely that their quantum states are left unchanged by ambient isotopy
of their world line
s. It is hoped that they can be used to make a quantum computer
resistant to decoherence. Since the world lines form a mathematical braid, braid theory
, a related field to knot theory
, is used in studying the properties of such a computer, called a topological quantum computer
.
Climbing
Climbing is the activity of using one's hands and feet to ascend a steep object. It is done both for recreation and professionally, as part of activities such as maintenance of a structure, or military operations.Climbing activities include:* Bouldering: Ascending boulders or small...
or sailing
Sailing
Sailing is the propulsion of a vehicle and the control of its movement with large foils called sails. By changing the rigging, rudder, and sometimes the keel or centre board, a sailor manages the force of the wind on the sails in order to move the boat relative to its surrounding medium and...
. Knots were also regarded as having spiritual and religious symbolism in addition to their aesthetic qualities. The endless knot
Endless knot
The endless knot or eternal knot is a symbolic knot and one of the Eight Auspicious Symbols. It is an important cultural marker in places significantly influenced by Tibetan Buddhism such as Tibet, Mongolia, Tuva, Kalmykia, and Buryatia...
appears in Tibetan Buddhism, while the Borromean rings
Borromean rings
In mathematics, the Borromean rings consist of three topological circles which are linked and form a Brunnian link, i.e., removing any ring results in two unlinked rings.- Mathematical properties :...
have made repeated appearances in different cultures, often symbolizing unity. The Celtic
Celtic Christianity
Celtic Christianity or Insular Christianity refers broadly to certain features of Christianity that were common, or held to be common, across the Celtic-speaking world during the Early Middle Ages...
monks who created the Book of Kells
Book of Kells
The Book of Kells is an illuminated manuscript Gospel book in Latin, containing the four Gospels of the New Testament together with various prefatory texts and tables. It was created by Celtic monks ca. 800 or slightly earlier...
lavished entire pages with intricate Celtic knot
Celtic knot
Celtic knots are a variety of knots and stylized graphical representations of knots used for decoration, used extensively in the Celtic style of Insular art. These knots are most known for their adaptation for use in the ornamentation of Christian monuments and manuscripts, such as the 8th-century...
work.
Knots were studied from a mathematical viewpoint by Carl Friedrich Gauss
Carl Friedrich Gauss
Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...
, who in 1833 developed the Gauss linking integral for computing the linking number
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...
of two knots. His student Johann Benedict Listing
Johann Benedict Listing
Johann Benedict Listing was a German mathematician.J. B. Listing was born in Frankfurt and died in Göttingen. He first introduced the term "topology", in a famous article published in 1847, although he had used the term in correspondence some years earlier...
, after whom Listing's knot
Figure-eight knot (mathematics)
In knot theory, a figure-eight knot is the unique knot with a crossing number of four. This is the smallest possible crossing number except for the unknot and trefoil knot...
is named, furthered their study.
Knot theory
In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical language, a knot is an embedding of a...
would arrive later with Sir William Thomson (Lord Kelvin) and his theory of vortex atoms.
In 1867 after observing Scottish
Scotland
Scotland is a country that is part of the United Kingdom. Occupying the northern third of the island of Great Britain, it shares a border with England to the south and is bounded by the North Sea to the east, the Atlantic Ocean to the north and west, and the North Channel and Irish Sea to the...
physicist
Physicist
A physicist is a scientist who studies or practices physics. Physicists study a wide range of physical phenomena in many branches of physics spanning all length scales: from sub-atomic particles of which all ordinary matter is made to the behavior of the material Universe as a whole...
Peter Tait
Peter Guthrie Tait
Peter Guthrie Tait FRSE was a Scottish mathematical physicist, best known for the seminal energy physics textbook Treatise on Natural Philosophy, which he co-wrote with Kelvin, and his early investigations into knot theory, which contributed to the eventual formation of topology as a mathematical...
's experiments involving smoke rings, Thomson came to the idea that atoms were knots of swirling vortices in the æther
Luminiferous aether
In the late 19th century, luminiferous aether or ether, meaning light-bearing aether, was the term used to describe a medium for the propagation of light....
. 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
Spectroscopy
Spectroscopy is the study of the interaction between matter and radiated energy. Historically, spectroscopy originated through the study of visible light dispersed according to its wavelength, e.g., by a prism. Later the concept was expanded greatly to comprise any interaction with radiative...
light at only the discrete wavelength
Wavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...
s that they do. For example, Thomson thought that sodium could be the Hopf link
Hopf link
thumb|right|[[Skein relation]] for the Hopf link.In mathematical knot theory, the Hopf link, named after Heinz Hopf, is the simplest nontrivial link with more than one component. It consists of two circles linked together exactly once...
due to its two lines of spectra.
Tait subsequently began listing unique knots in the belief that he was creating a table of elements. He formulated what are now known as the Tait conjectures
Tait conjectures
The Tait conjectures are conjectures made by Peter Guthrie Tait in his study of knots. The Tait conjectures involve concepts in knot theory such as alternating knots, chirality, and writhe...
on alternating knot
Alternating knot
In knot theory, a link diagram is alternating if the crossings alternate under, over, under, over, as you travel along each component of the link. A link is alternating if it has an alternating diagram....
s. (The conjectures were finally resolved in the 1990s.) Tait's knot tables were subsequently improved upon by C. N. Little and T. P. Kirkman.
James Clerk Maxwell
James Clerk Maxwell
James Clerk Maxwell of Glenlair was a Scottish physicist and mathematician. His most prominent achievement was formulating classical electromagnetic theory. This united all previously unrelated observations, experiments and equations of electricity, magnetism and optics into a consistent theory...
, 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' 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 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
Vortex theory
Vortex theory may refer to:*Mechanical explanations of gravitation — a theory to explain gravitation.*History of knot theory — a theory to explain the atom....
became completely obsolete, and knot theory ceased to be of great scientific interest. Modern physics demonstrates that the discrete wavelengths depend on quantum energy levels.
Following the development of topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
in the early 20th century spearheaded by Henri Poincaré
Henri Poincaré
Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...
, topologists such as Max Dehn
Max Dehn
Max Dehn was a German American mathematician and a student of David Hilbert. He is most famous for his work in geometry, topology and geometric group theory...
, J. W. Alexander
J. W. Alexander
J. W. Alexander may refer to:* James Waddel Alexander , American Presbyterian minister and author* John White Alexander , American portrait painter and illustrator...
, and Kurt Reidemeister
Kurt Reidemeister
Kurt Werner Friedrich Reidemeister was a mathematician born in Braunschweig , Germany.He received his doctorate in 1921 with a thesis in algebraic number theory at the University of Hamburg under the supervision of Erich Hecke. In 1923 he was appointed assistant professor at the University of Vienna...
, investigated knots. Out of this sprang the Reidemeister move
Reidemeister move
In the mathematical area of knot theory, a Reidemeister move refers to one of three local moves on a link diagram. In 1926, Kurt Reidemeister and independently, in 1927, J.W. Alexander and G.B...
s and the Alexander polynomial
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...
. Dehn also developed Dehn surgery
Dehn surgery
In topology, a branch of mathematics, a Dehn surgery, named after Max Dehn, is a specific construction used to modify 3-manifolds. The process takes as input a 3-manifold together with a link...
, which related knots to the general theory of 3-manifolds, and formulated the Dehn problems in group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
, such as the word problem
Word problem
The term word problem has several meanings:* word problem is a type of textbook problem designed to help students apply abstract mathematical concepts to "real-world" situations...
. Early pioneers in the first half of the 20th century include Ralph Fox
Ralph Fox
Ralph Hartzler Fox 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 an important role in the modernization and main-streaming of knot theory.Ralph Fox attended Swarthmore...
, who popularized the subject. In this early period, knot theory primarily consisted of study into the knot group
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....
and homological
Homology theory
In mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces. It can be broadly defined as the study of homology theories on topological spaces.-The general idea:...
invariants of the knot complement
Knot complement
In mathematics, the knot complement of a tame knot K is the complement of the interior of the embedding of a solid torus into the 3-sphere. To make this precise, suppose that K is a knot in a three-manifold M. Let N be a thickened neighborhood of K; so N is a solid torus...
.
In 1961 Wolfgang Haken
Wolfgang Haken
Wolfgang Haken is a mathematician who specializes in topology, in particular 3-manifolds.In 1976 together with colleague Kenneth Appel at the University of Illinois at Urbana-Champaign, Haken solved one of the most famous problems in mathematics, the four-color theorem...
discovered an algorithm that can determine whether or not a knot is a knot
Unknotting problem
In mathematics, the unknotting problem is the problem of algorithmically recognizing the unknot, given some representation of a knot, e.g., a knot diagram. There are several types of unknotting algorithms...
. 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
Friedhelm Waldhausen is a German mathematician known for his work in algebraic topology.-Academic life:...
announced the completion of Haken's program based on his results and those of Klaus Johannson, William Jaco
William Jaco
William "Bus" H. Jaco is the Grace B. Kerr Professor of mathematics at Oklahoma State University. He works on 3-manifolds and discovered the JSJ decomposition....
, Peter Shalen
Peter Shalen
Peter B. Shalen 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 College in 1966 and his Ph.D. from Harvard University in 1972...
, 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 Thurston
William Paul Thurston is an American mathematician. He is a pioneer in the field of low-dimensional topology. In 1982, he was awarded the Fields Medal for his contributions to the study of 3-manifolds...
's hyperbolization theorem
Geometrization conjecture
Thurston's geometrization conjecture states that compact 3-manifolds can be decomposed canonically into submanifolds that have geometric structures. The geometrization conjecture is an analogue for 3-manifolds of the uniformization theorem for surfaces...
introduced the theory of hyperbolic 3-manifold
Hyperbolic 3-manifold
A hyperbolic 3-manifold is a 3-manifold equipped with a complete Riemannian metric of constant sectional curvature -1. In other words, it is the quotient of three-dimensional hyperbolic space by a subgroup of hyperbolic isometries acting freely and properly discontinuously...
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
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
and algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
. Important results followed, including the Gordon-Luecke theorem
Gordon-Luecke theorem
In mathematics, the Gordon–Luecke theorem on knot complements states that every homeomorphism between two complements of knots in the 3-sphere extends to give a self-homeomorphism of the 3-sphere...
, which showed that knots were determined (up to mirror-reflection) by their complements, and the Smith conjecture
Smith conjecture
In mathematics, the Smith conjecture states that if f is a diffeomorphism of the 3-sphere, of finite order then the fixed point set of f cannot be a nontrivial knot....
.
Interest in knot theory from the general mathematical community grew significantly after Vaughan Jones
Vaughan Jones
Sir Vaughan Frederick Randal Jones, KNZM, FRS, FRSNZ is a New Zealand mathematician, known for his work on von Neumann algebras, knot polynomials and conformal field theory. He was awarded a Fields Medal in 1990, and famously wore a New Zealand rugby jersey when he accepted the prize...
' discovery of the Jones polynomial in 1984. This led to other knot polynomials such as the bracket polynomial
Bracket polynomial
In the mathematical field of knot theory, the bracket polynomial is a polynomial invariant of framed links. Although it is not an invariant of knots or links , a suitably "normalized" version yields the famous knot invariant called the Jones polynomial...
, HOMFLY polynomial
HOMFLY polynomial
In the mathematical field of knot theory, the HOMFLY polynomial, sometimes called the HOMFLY-PT polynomial or 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 l....
, and Kauffman polynomial. Jones was awarded the Fields medal
Fields Medal
The Fields Medal, officially known as International Medal for Outstanding Discoveries in Mathematics, is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union , a meeting that takes place every four...
in 1990 for this work. In 1988 Edward Witten
Edward Witten
Edward Witten is an American theoretical physicist with a focus on mathematical physics who is currently a professor of Mathematical Physics at the Institute for Advanced Study....
proposed a new framework for the Jones polynomial, utilizing existing ideas from mathematical physics
Mathematical physics
Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...
, such as Feynman path integrals, and introducing new notions such as topological quantum field theory
Topological quantum field theory
A topological quantum field theory is a quantum field theory which computes topological invariants....
. 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
3-manifold
In mathematics, a 3-manifold is a 3-dimensional manifold. The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds.Phenomena in three dimensions...
s. Simultaneous, but different, approaches by other mathematicians resulted in the Witten-Reshetikhin-Turaev invariants and various so-called "quantum invariant
Quantum invariant
In the mathematical field of knot theory, a 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*Kontsevich invariant*Kashaev's invariant...
s", which appear to be the mathematically rigorous version of Witten's invariants .
In 1992, the Journal of Knot Theory and Its Ramifications
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 creating connections between with other branches of...
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
Finite type invariant
In the mathematical theory of knots, a finite type invariant is a knot invariant that can be extended to an invariant of certain singular knots that vanishes on singular knots with m + 1 singularities and does not vanish on some singular knot with 'm' singularities...
s, were discovered by Vassiliev and Goussarov
Mikhail Goussarov
Mikhail Goussarov was a Soviet mathematician who worked in low-dimensional topology. He and Victor Vassiliev independently discovered finite type invariants of knots and links. He accidentally drowned at the age of 41 in Tel Aviv, Israel.- See also :* maintained by Dror Bar-Natan....
. These invariants, initially described using "classical" topological means, were shown by 1994 Fields Medalist Maxim Kontsevich
Maxim Kontsevich
Maxim Lvovich Kontsevich is a Russian mathematician. He is a professor at the Institut des Hautes Études Scientifiques and a distinguished professor at the University of Miami...
to result from integration
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
, using the Kontsevich integral, of certain algebraic structures .
These breakthroughs were followed by the discovery of Khovanov homology
Khovanov homology
In mathematics, Khovanov homology is an invariant of oriented knots and links that arises as the homology of a chain complex. It may be regarded as a categorification of the Jones polynomial....
and knot Floer homology
Floer homology
Floer homology is a mathematical tool used in the study of symplectic geometry and low-dimensional topology. First introduced by Andreas Floer in his proof of the Arnold conjecture in symplectic geometry, Floer homology is a novel homology theory arising as an infinite dimensional analog of finite...
, 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
Biology is a natural science concerned with the study of life and living organisms, including their structure, function, growth, origin, evolution, distribution, and taxonomy. Biology is a vast subject containing many subdivisions, topics, and disciplines...
and chemistry
Chemistry
Chemistry is the science of matter, especially its chemical reactions, but also its composition, structure and properties. Chemistry is concerned with atoms and their interactions with other atoms, and particularly with the properties of chemical bonds....
. Knot theory can be used to determine if a molecule is chiral
Chirality (chemistry)
A chiral molecule is a type of molecule that lacks an internal plane of symmetry and thus has a non-superimposable mirror image. The feature that is most often the cause of chirality in molecules is the presence of an asymmetric carbon atom....
(has a "handedness") or not. Chemical compounds of different handedness can have drastically differing properties, thalidomide being a notable example of this. More generally, knot theoretic methods have been used in studying topoisomer
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 catenanes...
s, topologically different arrangements of the same chemical formula. The closely related theory of tangles have been effectively used in studying the action of certain enzymes on DNA. The interdisciplinary field of physical knot theory
Physical knot theory
Physical knot theory is the study of mathematical models of knotting phenomena, often motivated by physical considerations from biology, chemistry, and physics. Traditional knot theory models a knot as a simple closed loop in three dimensional space. Such a knot has no thickness or physical...
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
Quasiparticle
In physics, quasiparticles are emergent phenomena that occur when a microscopically complicated system such as a solid behaves as if it contained different weakly interacting particles in free space...
s such as nonabelian anyon
Anyon
In physics, an anyon is a type of particle that occurs only in two-dimensional systems. It is a generalization of the fermion and boson concept.-From theory to reality:...
s exhibit useful topological properties, namely that their quantum states are left unchanged by ambient isotopy
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", a manifold, taking a submanifold to another submanifold. For example in knot theory, one considers two knots the same if one can distort one knot into the...
of their world line
World line
In physics, the world line of an object is the unique path of that object as it travels through 4-dimensional spacetime. The concept of "world line" is distinguished from the concept of "orbit" or "trajectory" by the time dimension, and typically encompasses a large area of spacetime wherein...
s. It is hoped that they can be used to make a quantum computer
Quantum computer
A quantum computer is a device for computation that makes direct use of quantum mechanical phenomena, such as superposition and entanglement, to perform operations on data. Quantum computers are different from traditional computers based on transistors...
resistant to decoherence. Since the world lines form a mathematical braid, braid theory
Braid theory
In topology, a branch of mathematics, braid theory is an abstract geometric theory studying the everyday braid concept, and some generalizations. The idea is that braids can be organized into groups, in which the group operation is 'do the first braid on a set of strings, and then follow it with a...
, a related field to knot theory
Knot theory
In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical language, a knot is an embedding of a...
, is used in studying the properties of such a computer, called a topological quantum computer
Topological quantum computer
A topological quantum computer is a theoretical quantum computer that employs two-dimensional quasiparticles called anyons, whose world lines cross over one another to form braids in a three-dimensional spacetime . These braids form the logic gates that make up the computer...
.
History
- 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