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Knots have been used for basic purposes such as recording information, fastening and tying objects together, for thousands of years. The early significant stimulus in knot theory would arrive later with Sir William Thomson (Lord Kelvin) and his vortex theory of the atom.
Different knots are 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 knotwork.
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.
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 wavelengths that they do. For example, Thomson thought that sodium could be the Hopf link due to its two lines of spectra. [1]
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 on alternating knots. (The conjectures were proved in the 1990s.) Tait's knot tables were subsequently improved upon by C. N. Little and Thomas Kirkman. [1] : 6
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'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 levels.
Following the development 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 moves and the Alexander polynomial. [1] : 15–45 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 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 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-manifolds 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. [1] : 71–89 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. [2] Witten also received the Fields medal, in 1990, partly for this work. Witten's description of the Jones polynomial implied related invariants for 3-manifolds. Simultaneous, but different, approaches by other mathematicians resulted in the Witten–Reshetikhin–Turaev invariants and various so-called "quantum invariants", which appear to be the mathematically rigorous version of Witten's invariants. [3] In the 1980s John Horton Conway discovered a procedure for unknotting knots gradually known as Conway notation.
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 invariants, 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. [4]
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 topoisomers, 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. [5] 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 quasiparticles such as nonabelian anyons exhibit useful topological properties, namely that their quantum states are left unchanged by ambient isotopy of their world lines. 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. [6]
A development related and complementary to knot theory is circuit topology which was originally proposed by Alireza Mashaghi, [7] as a theory that focuses on open chains that include intra chain contacts or bonds. The theory was historically developed to address problems in molecular topology and in particular in biology. [8]
Edward Witten 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, quantum gravity, supersymmetric quantum field theories, and other areas of mathematical physics. Witten's work has also significantly impacted pure mathematics. In 1990, he became the first physicist to be awarded a Fields Medal by the International Mathematical Union, for his mathematical insights in physics, such as his 1981 proof of the positive energy theorem in general relativity, and his interpretation of the Jones invariants of knots as Feynman integrals. He is considered the practical founder of M-theory.
Maxim Lvovich Kontsevich is a Russian and French mathematician and mathematical physicist. He is a professor at the Institut des Hautes Études Scientifiques and a distinguished professor at the University of Miami. He received the Henri Poincaré Prize in 1997, the Fields Medal in 1998, the Crafoord Prize in 2008, the Shaw Prize and Breakthrough Prize in Fundamental Physics in 2012, and the Breakthrough Prize in Mathematics in 2015.
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. In mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, . Two mathematical knots are equivalent if one can be transformed into the other via a deformation of upon itself ; these transformations correspond to manipulations of a knotted string that do not involve cutting it or passing it through itself.
In the mathematical field of knot theory, a knot invariant is a quantity (in a broad sense) defined for each knot which is the same for equivalent knots. The equivalence is often given by ambient isotopy but can be given by homeomorphism. Some invariants are indeed numbers (algebraic), but invariants can range from the simple, such as a yes/no answer, to those as complex as a homology theory (for example, "a knot invariant is a rule that assigns to any knot K a quantity φ(K) such that if K and K' are equivalent then φ(K) = φ(K')."). Research on invariants is not only motivated by the basic problem of distinguishing one knot from another but also to understand fundamental properties of knots and their relations to other branches of mathematics. Knot invariants are thus used in knot classification, both in "enumeration" and "duplication removal".
A knot invariant is a quantity defined on the set of all knots, which takes the same value for any two equivalent knots. For example, a knot group is a knot invariant.
Typically a knot invariant is a combinatorial quantity defined on knot diagrams. Thus if two knot diagrams differ with respect to some knot invariant, they must represent different knots. However, as is generally the case with topological invariants, if two knot diagrams share the same values with respect to a [single] knot invariant, then we still cannot conclude that the knots are the same.
The Chern–Simons theory is a 3-dimensional topological quantum field theory of Schwarz type developed by Edward Witten. It was discovered first by mathematical physicist Albert Schwarz. It is named after mathematicians Shiing-Shen Chern and James Harris Simons, who introduced the Chern–Simons 3-form. In the Chern–Simons theory, the action is proportional to the integral of the Chern–Simons 3-form.
In gauge theory and mathematical physics, a topological quantum field theory is a quantum field theory which computes topological invariants.
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 Euclidean space, the linking number is always an integer, but may be positive or negative depending on the orientation of the two curves.
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 polynomial in the variable with integer coefficients.
Algorithmic topology, or computational topology, is a subfield of topology with an overlap with areas of computer science, in particular, computational geometry and computational complexity theory.
In the mathematical theory of knots, a finite type invariant, or Vassiliev 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. It is then said to be of type or order m.
In knot theory, a virtual knot is a generalization of knots in 3-dimensional Euclidean space, R3, to knots in thickened surfaces modulo an equivalence relation called stabilization/destabilization. Here is required to be closed and oriented. Virtual knots were first introduced by Kauffman (1999).
Louis Hirsch Kauffman is an American mathematician, mathematical physicist, and professor of mathematics in the Department of Mathematics, Statistics, and Computer science at the University of Illinois at Chicago. He does research in topology, knot theory, topological quantum field theory, quantum information theory, and diagrammatic and categorical mathematics. He is best known for the introduction and development of the bracket polynomial and the Kauffman polynomial.
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, and conversely any finite type invariant can be presented as a linear combination of such coefficients. It was defined by Maxim Kontsevich.
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.
Vladimir Georgievich Turaev is a Russian mathematician, specializing in topology.
Efstratia (Effie) Kalfagianni is a Greek American mathematician specializing in low-dimensional topology.
Serguei Barannikov is a mathematician, known for his works in algebraic topology, algebraic geometry and mathematical physics.
In the mathematical field of quantum topology, the Reshetikhin–Turaev invariants (RT-invariants) are a family of quantum invariants of framed links. Such invariants of framed links also give rise to invariants of 3-manifolds via the Dehn surgery construction. These invariants were discovered by Nicolai Reshetikhin and Vladimir Turaev in 1991, and were meant to be a mathematical realization of Witten's proposed invariants of links and 3-manifolds using quantum field theory.
