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Bing Double
In knot theory, a field of mathematics, the Bing double of a knot is a link with two components which follow the pattern of the knot and "hook together". Bing doubles were introduced in by their namesake, the American mathematician R. H. Bing. The Bing double of a slice knot is a slice link, though it is unknown whether the converse is true. The components of a Bing double bound disjoint Seifert surfaces. The Bing double of a knot is defined by placing the Bing double of the unknot in the solid torus surrounding it, as shown in the figure, and then twisting that solid torus into the shape of . This definition is similar to that for Whitehead doubles. The Bing double of the unknot is also called the Bing link. See also * Slice knot * Seifert surface * Satellite knot * Whitehead double References Notes Sources *. *. *. Further reading *. *. Knot theory Knot theory stubs {{knottheory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bing Double Of Unknot
Bing most often refers to: * Bing Crosby (1903–1977), American singer * Microsoft Bing, a web search engine Bing may also refer to: Food and drink * Bing (bread), a Chinese flatbread * Bing (soft drink), a UK brand * Bing cherry, a variety of cherry * Twin Bing or Bing, a candy made by Palmer Candy Company Names * Bing (German surname), a German-language surname * Bing (Chinese surname) (邴), a Chinese surname Places * Bing Prefecture, an ancient Chinese province * Bing, Hormozgan, a village in Hormozgan Province, Iran * Binag, Sistan and Baluchestan, a village in Sistan and Baluchestan Province, Iran * Manor of Byng, Suffolk, England Television * Bing (TV series), ''Bing'' (TV series), a British children's television series * Bada Bing or the Bing, a fictional strip club in ''The Sopranos'' * Bing or Evan Chambers, a List of Greek characters#Main characters, character in ''Greek'' Other uses * Bing (company), a German company that manufactured toys and kitchen utensils ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knot Theory
In 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, 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. 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 it through itself. Knots can be described in various ways. Using different description methods, there may be more than one description of the same knot. For example, a common method of describing a knot is a planar diagram called a knot diagram, in which any knot can be drawn in many different ways. Therefore, a fundamental p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Slice Knot
A slice knot is a knot (mathematics), mathematical knot in 3-dimensional space that bounds an embedded disk in 4-dimensional space. Definition A knot K \subset S^3 is said to be a topologically slice knot or a smoothly slice knot, if it is the boundary of an Embedding, embedded disk in the 4-ball B^4, which is Local flatness, locally flat or Smoothness, smooth, respectively. Here we use S^3 = \partial B^4: the 3-sphere S^3 = \ is the boundary (topology), boundary of the four-dimensional ball (mathematics), ball B^4 = \. Every smoothly slice knot is topologically slice because a smoothly embedded disk is locally flat. Usually, smoothly slice knots are also just called slice. Both types of slice knots are important in 3- and 4-dimensional topology. Smoothly slice knots are often illustrated using knots diagrams of ribbon knots and it is an open question whether there are any smoothly slice knots which are not ribbon knots (′Slice-ribbon conjecture′). Cone construction The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Slice Link
In mathematics, two links L_0 \subset S^n and L_1 \subset S^n are concordant if there exists an embedding f : L_0 \times ,1\to S^n \times ,1/math> such that f(L_0 \times \) = L_0 \times \ and f(L_0 \times \) = L_1 \times \. By its nature, link concordance is an equivalence relation. It is weaker than isotopy, and stronger than homotopy: isotopy implies concordance implies homotopy. A link is a slice link if it is concordant to the unlink. Concordance invariants A function of a link that is invariant under concordance is called a concordance invariant. The linking number of any two components of a link is one of the most elementary concordance invariants. The signature of a knot is also a concordance invariant. A subtler concordance invariant are the Milnor invariants, and in fact all rational finite type concordance invariants are Milnor invariants and their products, though non-finite type concordance invariants exist. Higher dimensions One can analogously define con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Seifert Surface
In mathematics, a Seifert surface (named after German mathematician Herbert Seifert) is an orientable surface whose boundary is a given knot or link. Such surfaces can be used to study the properties of the associated knot or link. For example, many knot invariants are most easily calculated using a Seifert surface. Seifert surfaces are also interesting in their own right, and the subject of considerable research. Specifically, let ''L'' be a tame oriented knot or link in Euclidean 3-space (or in the 3-sphere). A Seifert surface is a compact, connected, oriented surface ''S'' embedded in 3-space whose boundary is ''L'' such that the orientation on ''L'' is just the induced orientation from ''S''. Note that any compact, connected, oriented surface with nonempty boundary in Euclidean 3-space is the Seifert surface associated to its boundary link. A single knot or link can have many different inequivalent Seifert surfaces. A Seifert surface must be oriented. It is possi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Solid Torus Surrounding Bing Double Of Unknot
Solid is a state of matter where molecules are closely packed and can not slide past each other. Solids resist compression, expansion, or external forces that would alter its shape, with the degree to which they are resisted dependent upon the specific material under consideration. Solids also always possess the least amount of kinetic energy per atom/molecule relative to other phases or, equivalently stated, solids are formed when matter in the liquid / gas phase is cooled below a certain temperature. This temperature is called the melting point of that substance and is an intrinsic property, i.e. independent of how much of the matter there is. All matter in solids can be arranged on a microscopic scale under certain conditions. Solids are characterized by structural rigidity and resistance to applied external forces and pressure. Unlike liquids, solids do not flow to take on the shape of their container, nor do they expand to fill the entire available volume like a gas. Much ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Solid Torus
In mathematics, a solid torus is the topological space formed by sweeping a disk around a circle. It is homeomorphic to the Cartesian product S^1 \times D^2 of the disk and the circle, endowed with the product topology. A standard way to visualize a solid torus is as a toroid, embedded in 3-space. However, it should be distinguished from a torus, which has the same visual appearance: the torus is the two-dimensional space on the boundary of a toroid, while the solid torus includes also the compact interior space enclosed by the torus. A solid torus is a torus plus the volume inside the torus. Real-world objects that approximate a ''solid torus'' include O-rings, non-inflatable lifebuoys, ring doughnuts, and bagels. Topological properties The solid torus is a connected, compact, orientable 3-dimensional manifold with boundary. The boundary is homeomorphic to S^1 \times S^1, the ordinary torus. Since the disk D^2 is contractible, the solid torus has the homotopy type o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Whitehead Double
Whitehead may refer to: * Whitehead (comedo), a blocked sweat/sebaceous duct of the skin * Whitehead (bird), a small species of passerine bird, endemic to New Zealand * Whitehead building, heritage listed residence of the principal of the University of Adelaide's Lincoln College * Whitehead (patience), a patience game related to Klondike * Whitehead (surname) * Whitehead & Co., a former torpedo company founded by Robert Whitehead in 1875 * Whitehead SpA, one of the names of the later torpedo factory in Livorno * Whiteheads, another name for the wheat disease take-all * USS ''Whitehead'' (1861–1865), American Civil War, 136-ton screw steam gunboat Places * Canada: ** Rural Municipality of Whitehead, Manitoba ** Whitehead, Nova Scotia, on Tor Bay * Hong Kong ** Whitehead, Hong Kong, a cape at Wu Kai Sha * Northern Ireland ** Whitehead, County Antrim, a small town in Northern Ireland * United States: ** Lake Whitehead, a reservoir in Napa County, California ** Whitehead, Miss ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Slice Knot
A slice knot is a knot (mathematics), mathematical knot in 3-dimensional space that bounds an embedded disk in 4-dimensional space. Definition A knot K \subset S^3 is said to be a topologically slice knot or a smoothly slice knot, if it is the boundary of an Embedding, embedded disk in the 4-ball B^4, which is Local flatness, locally flat or Smoothness, smooth, respectively. Here we use S^3 = \partial B^4: the 3-sphere S^3 = \ is the boundary (topology), boundary of the four-dimensional ball (mathematics), ball B^4 = \. Every smoothly slice knot is topologically slice because a smoothly embedded disk is locally flat. Usually, smoothly slice knots are also just called slice. Both types of slice knots are important in 3- and 4-dimensional topology. Smoothly slice knots are often illustrated using knots diagrams of ribbon knots and it is an open question whether there are any smoothly slice knots which are not ribbon knots (′Slice-ribbon conjecture′). Cone construction The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Satellite Knot
In the mathematical theory of knots, a satellite knot is a knot that contains an incompressible, non boundary-parallel torus in its complement. Every knot is either hyperbolic, a torus, or a satellite knot. The class of satellite knots include composite knots, cable knots, and Whitehead doubles. A satellite ''link'' is one that orbits a companion knot ''K'' in the sense that it lies inside a regular neighborhood of the companion. A satellite knot K can be picturesquely described as follows: start by taking a nontrivial knot K' lying inside an unknotted solid torus V. Here "nontrivial" means that the knot K' is not allowed to sit inside of a 3-ball in V and K' is not allowed to be isotopic to the central core curve of the solid torus. Then tie up the solid torus into a nontrivial knot. This means there is a non-trivial embedding f\colon V \to S^3 and K = f\left(K'\right). The central core curve of the solid torus V is sent to a knot H, which is called the "companion knot" a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Proceedings Of The Cambridge Philosophical Society
''Mathematical Proceedings of the Cambridge Philosophical Society'' is a mathematical journal published by Cambridge University Press for the Cambridge Philosophical Society. It aims to publish original research papers from a wide range of pure and applied mathematics. The journal, titled ''Proceedings of the Cambridge Philosophical Society'' before 1975, has been published since 1843. Abstracting and indexing The journal is abstracted and indexed in *MathSciNet *Science Citation Index Expanded *Scopus *ZbMATH Open See also *Cambridge Philosophical Society The Cambridge Philosophical Society (CPS) is a scientific society at the University of Cambridge. It was founded in 1819. The name derives from the medieval use of the word philosophy to denote any research undertaken outside the fields of law ... External linksofficial website References Academic journals associated with learned and professional societies Cambridge University Press academic journals Mathematics e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
