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Knot Polynomials
In the mathematical field of knot theory, a knot polynomial is a knot invariant in the form of a polynomial whose coefficients encode some of the properties of a given knot. History The first knot polynomial, the Alexander polynomial, was introduced by James Waddell Alexander II in 1923. Other knot polynomials were not found until almost 60 years later. In the 1960s, John Conway came up with a skein relation for a version of the Alexander polynomial, usually referred to as the Alexander–Conway polynomial. The significance of this skein relation was not realized until the early 1980s, when Vaughan Jones discovered the Jones polynomial. This led to the discovery of more knot polynomials, such as the so-called HOMFLY polynomial. Soon after Jones' discovery, Louis Kauffman noticed the Jones polynomial could be computed by means of a partition function (state-sum model), which involved the bracket polynomial, an invariant of framed knots. This opened up avenues of research l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Skein (HOMFLY)
Skein may refer to: * A flock of geese or ducks in flight * A wound ball of yarn with a centre pull strand; see Hank * A metal piece fitted over the end of a wagon axle, to which the wheel is mounted * Skein (unit), a unit of length used by weavers and tailors * Skein dubh, a Scottish knife * Skein (fish), the egg sack of the fish eggs and/or the eggs themselves * Skein module, a mathematical concept * Skein relation, a mathematical concept often used to give a simple definition of knot polynomials * Skein (comics), a fictional supervillain in the Marvel Comics universe * Skein (hash function), a candidate hash function to the NIST hash function competition from Bruce Schneier et al. See also * ''The Tangled Skein'', a novel by Baroness Orczy * ''With a Tangled Skein'', a novel by Piers Anthony, book three of ''Incarnations of Immortality'' * Skien Skien () is a city and municipality in Vestfold og Telemark county in Norway. In modern times it is regarded as part of the tradition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Framed Knot
In mathematics, a knot is an embedding of the circle into three-dimensional Euclidean space, (also known as ). Often two knots are considered equivalent if they are ambient isotopic, that is, if there exists a continuous deformation of which takes one knot to the other. A crucial difference between the standard mathematical and conventional notions of a knot is that mathematical knots are closed — there are no ends to tie or untie on a mathematical knot. Physical properties such as friction and thickness also do not apply, although there are mathematical definitions of a knot that take such properties into account. The term ''knot'' is also applied to embeddings of in , especially in the case . The branch of mathematics that studies knots is known as knot theory and has many relations to graph theory. Formal definition A knot is an embedding of the circle () into three-dimensional Euclidean space (), or the 3-sphere (), since the 3-sphere is compact. Two knots are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unknot
In the mathematical theory of knots, the unknot, not knot, or trivial knot, is the least knotted of all knots. Intuitively, the unknot is a closed loop of rope without a knot tied into it, unknotted. To a knot theorist, an unknot is any embedded topological circle in the 3-sphere that is ambient isotopic (that is, deformable) to a geometrically round circle, the standard unknot. The unknot is the only knot that is the boundary of an embedded disk, which gives the characterization that only unknots have Seifert genus 0. Similarly, the unknot is the identity element with respect to the knot sum operation. Unknotting problem Deciding if a particular knot is the unknot was a major driving force behind knot invariants, since it was thought this approach would possibly give an efficient algorithm to recognize the unknot from some presentation such as a knot diagram. Unknot recognition is known to be in both NP and co-NP. It is known that knot Floer homology and Khova ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 joined so it cannot be undone, Unknot, the simplest knot being a ring (or "unknot"). In mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, \mathbb^3 (in topology, a circle is not bound to the classical geometric concept, but to all of its homeomorphisms). Two mathematical knots are equivalent if one can be transformed into the other via a deformation of \mathbb^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 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 descr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zoltán Szabó (mathematician)
Zoltán Szabó (born November 24, 1965) is a professor of mathematics at Princeton University known for his work on Heegaard Floer homology. Education and career Szabó received his B.A. from Eötvös Loránd University in Budapest, Hungary in 1990, and he received his Ph.D. from Rutgers University in 1994. Together with Peter Ozsváth, Szabó created Heegaard Floer homology, a homology theory for 3-manifolds. For this contribution to the field of topology, Ozsváth and Szabó were awarded the 2007 Oswald Veblen Prize in Geometry. In 2010, he was elected honorary member of the Hungarian Academy of Sciences The Hungarian Academy of Sciences ( hu, Magyar Tudományos Akadémia, MTA) is the most important and prestigious learned society of Hungary. Its seat is at the bank of the Danube in Budapest, between Széchenyi rakpart and Akadémia utca. Its ma .... Selected publications *. *.Homology for Knots and Links'' American Mathematical Society, (2015) References External ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peter Ozsváth
Peter Steven Ozsváth (born October 20, 1967) is a professor of mathematics at Princeton University. He created, along with Zoltán Szabó, Heegaard Floer homology, a homology theory for 3-manifolds. Education Ozsváth received his Ph.D. from Princeton in 1994 under the supervision of John Morgan; his dissertation was entitled ''On Blowup Formulas For SU(2) Donaldson Polynomials''. Awards In 2007, Ozsváth was one of the recipients of the Oswald Veblen Prize in Geometry. In 2008 he was named a Guggenheim Fellow. In July 2017, he was a plenary lecturer in the Mathematical Congress of the Americas. He was elected a member of the National Academy of Sciences The National Academy of Sciences (NAS) is a United States nonprofit, non-governmental organization. NAS is part of the National Academies of Sciences, Engineering, and Medicine, along with the National Academy of Engineering (NAE) and the Nati ... in 2018. Selected publications * *Homology for Knots and Links'' Americ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heegaard Floer Homology
In mathematics, Floer homology is a tool for studying symplectic geometry and low-dimensional topology. Floer homology is a novel invariant that arises as an infinite-dimensional analogue of finite-dimensional Morse homology. Andreas Floer introduced the first version of Floer homology, now called Lagrangian Floer homology, in his proof of the Arnold conjecture in symplectic geometry. Floer also developed a closely related theory for Lagrangian submanifolds of a symplectic manifold. A third construction, also due to Floer, associates homology groups to closed three-dimensional manifolds using the Yang–Mills functional. These constructions and their descendants play a fundamental role in current investigations into the topology of symplectic and contact manifolds as well as (smooth) three- and four-dimensional manifolds. Floer homology is typically defined by associating to the object of interest an infinite-dimensional manifold and a real valued function on it. In the sympl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler Characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent. It is commonly denoted by \chi ( Greek lower-case letter chi). The Euler characteristic was originally defined for polyhedra and used to prove various theorems about them, including the classification of the Platonic solids. It was stated for Platonic solids in 1537 in an unpublished manuscript by Francesco Maurolico. Leonhard Euler, for whom the concept is named, introduced it for convex polyhedra more generally but failed to rigorously prove that it is an invariant. In modern mathematics, the Euler characteristic arises from homology and, more abstractly, homological algebra. Polyhedra The Euler characteristic \chi was classically defined for the surfaces of polyhedra, acc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Floer Homology
In mathematics, Floer homology is a tool for studying symplectic geometry and low-dimensional topology. Floer homology is a novel invariant that arises as an infinite-dimensional analogue of finite-dimensional Morse homology. Andreas Floer introduced the first version of Floer homology, now called Lagrangian Floer homology, in his proof of the Arnold conjecture in symplectic geometry. Floer also developed a closely related theory for Lagrangian submanifolds of a symplectic manifold. A third construction, also due to Floer, associates homology groups to closed three-dimensional manifolds using the Yang–Mills functional. These constructions and their descendants play a fundamental role in current investigations into the topology of symplectic and contact manifolds as well as (smooth) three- and four-dimensional manifolds. Floer homology is typically defined by associating to the object of interest an infinite-dimensional manifold and a real valued function on it. In the sympl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 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. We give the combinatorial definition of finite type invariant due to Goussarov, and (independently) Joan Birman and Xiao-Song Lin. Let ''V'' be a knot invariant. Define ''V''1 to be defined on a knot with one transverse singularity. Consider a knot ''K'' to be a smooth embedding of a circle into \R^3. Let ''K be a smooth immersion of a circle into \mathbb R^3 with one transverse double point. Then : V^1(K') = V(K_+) - V(K_-), where K_+ is obtained from ''K'' by resolving the double point by pushing up one strand above the other, and ''K_-'' is obta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mikhail Goussarov
Mikhail Goussarov (March 8, 1958, Leningrad – June 25, 1999, Tel Aviv) was a Soviet mathematician who worked in low-dimensional topology. He and Victor Vassiliev independently discovered finite type invariants of knots and links. He drowned at the age of 41 in an accident in Tel Aviv, Israel Israel (; he, יִשְׂרָאֵל, ; ar, إِسْرَائِيل, ), officially the State of Israel ( he, מְדִינַת יִשְׂרָאֵל, label=none, translit=Medīnat Yīsrāʾēl; ), is a country in Western Asia. It is situated .... See also Memorial pagemaintained by Dror Bar-Natan. 1958 births 1999 deaths 20th-century Russian mathematicians Deaths by drowning Accidental deaths in Israel {{Russia-mathematician-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Victor Anatolyevich Vasilyev
Victor Anatolyevich Vassiliev or Vasilyev ( ru , Виктор Анатольевич Васильев; born April 10, 1956), is a Soviet and Russian mathematician. He is best known for his discovery of the Vassiliev invariants in knot theory (also known as finite type invariants), which subsume many previously discovered polynomial knot invariants such as the Jones polynomial. He also works on singularity theory, topology, computational complexity theory, integral geometry, symplectic geometry, partial differential equations (geometry of wavefronts), complex analysis, combinatorics, and Picard–Lefschetz theory. Biography Vassiliev studied at the Faculty of Mathematics and Mechanics at the Lomonosov University in Moscow until 1981. From 1981 to 1987 he was Senior Researcher at the Documents and Archives Research Institute, Moscow and a part-time mathematics teacher at Specialized Mathematical School No. 57, Moscow. In 1982 he defended his Kandidat nauk thesis under Vladimir A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
